SOLUTION OF MONOTONE COMPLEMENTARITY AND GENERAL CONVEX PROGRAMMING...

SOLUTION OF MONOTONE COMPLEMENTARITY ANDGENERAL CONVEX PROGRAMMING PROBLEMS USING A

MODIFIED POTENTIAL REDUCTION INTERIOR POINT METHOD∗

KUO-LING HUANG AND SANJAY MEHROTRA†

Abstract. We present a homogeneous algorithm equipped with a modified potential functionfor the monotone complementarity problem. We show that this potential function is reduced by atleast a constant amount if a scaled Lipschitz condition is satisfied. A practical algorithm based onthis potential function is implemented in a software package named iOptimize. The implementa-tion in iOptimize maintains global linear polynomial-time convergence properties while achievingpractical performance. When compared with a mature software package MOSEK (barrier solver ver-sion 6.0.0.106), iOptimize solves convex quadratic programming problems, convex quadraticallyconstrained quadratic programming problems, and general convex programming problems in feweriterations. Moreover, several problems for which MOSEK fails are solved to optimality. We also findthat iOptimize seems to detect infeasibility more reliably than general nonlinear solvers Ipopt (ver-sion 3.9.2) and Knitro (version 8.0).

Key words. quadratic programs, quadratically constrained quadratic programs, convex pro-grams, homogeneous algorithms, interior point methods

AMS subject classifications. 90C06, 90C20, 90C25, 90C30, 90C33, 90C51, 49M15

1. Introduction. In this paper we consider a monotone complementarity prob-lem (MCP) of the following form

s = f(x)(1.1)

0 ≤ s ⊥ x ≥ 0,(1.2)

where x, s ∈ Rn, f(x) is a continuous monotone mapping fromRn+ := x ∈ Rn | x ≥ 0to Rn, and the notation 0 ≤ s ⊥ x ≥ 0 means that (x, s) ≥ 0 and that xT s = 0.The requirement xT s = 0 is called complementarity condition. Then, for everyx1, x2 ∈ Rn+, we have

(x1 − x2)T (f(x1)− f(x2)) ≥ 0.(1.3)

Let ∇f(x) denote the Jacobian matrix of f(x). If ∇f(x) is positive semidefinite forall x > 0, that is,

hT∇f(x)h ≥ 0,∀x > 0, h ∈ Rn,(1.4)

then f(x) is a continuous monotone mapping.(x, s) ⊂ R2n

+ is called a feasible solution of MCP if (1.1) is satisfied. It is calledan optimal or complementary solution if both (1.1) and (1.2) are satisfied. If MCPhas a complementary solution (x∗, s∗), then it has a maximal complementary solutionwhere the number of positive components of (x∗, s∗) is maximal.

Given an initial point (x0, s0) ⊂ R2n++ := x ∈ Rn | x > 0 , we would like to

compute a bounded sequence 〈xk, sk〉 ⊂ R2n++, k = 0, 1, . . . , such that

limk→∞

sk − f(xk)→ 0, and limk→∞

(xk)T sk → 0.(1.5)

∗The research of both authors was partially supported by grant DOE-SP0011568 and ONRN00014-09-10518, N00014-210051†Dept. of Industrial Engineering and Management Sciences, Northwestern University, Evanston,

IL, 60208, Email: [email protected]

1

2 K.-L. HUANG AND S. MEHROTRA

Let X := diag(x1, . . . , xn), and let

νf (·) : (0, 1) 7→ (1,∞)

be a monotone increasing function. We say that f satisfies a scaled Lipschitz condition(SLC) if

‖X(f(x+ h)− f(x)−∇f(x)h)‖1 ≤ νf (β)hT∇f(x)h,(1.6)

whenever

h ∈ Rn, x ∈ Rn++, ‖X−1h‖∞ ≤ β < 1,

where β ∈ (0, 1) is a constant. The analysis in this paper assumes that f satisfies SLC.Such a condition for MCP was used by Potra and Ye [31] to develop an interior pointmethod (IPM) for MCP. In their analysis Potra and Ye assumed that the set of strictlyfeasible points x ∈ Rn | x > 0, f(x) > 0 is nonempty, and a feasible starting pointx0 > 0 and s0 = f(x0) > 0 is available. Under these assumptions Potra and Ye [31]developed an IPM that achieves a constant reduction of the primal-dual potentialfunction

φρ(x, s) = ρ log xT s−n∑j=1

log xjsj ,(1.7)

for n+√n ≤ ρ ≤ 2n.

This type of potential function was introduced by Tanabe [35] and Todd andYe [36] for LP and further extended for linear complementarity problems by Kojimaet al. [22]. It satisfies the following inequality

(ρ− n) log xT s ≤ φρ(x, s)− n log n,

and hence it follows that

xT s ≤ ε whenever φρ(x, s) ≤ n log n− (ρ− n)| log ε|.

Therefore, starting from a feasible point, if this potential function is reduced by at leasta positive constant at each interior point iteration, then we will have xT s ≤ ε after atmost O(φρ(x

0, s0)−n log n+(ρ−n) |log ε|) iterations, where ε is the complementarityerror.

However, in practice an initial feasible point x0 for optimization problems may notbe available. To handle this issue, Ye et al. [39] developed a homogeneous IPM for lin-ear programming (LP) that can start from any (infeasible) interior point and achievesthe best known convergence complexity without introducing a big M constant. More-over, it detects the infeasibility of problems unambiguously. Xu et al. [38] developeda simplified (and equivalent) version of the homogeneous IPM, and showed that withsimilar computational efforts, their method can be more reliable than conventionalprimal-dual IPM while solving infeasible problems in Netlib [8] LP benchmark li-brary.

Andersen et al. [10] implemented a homogeneous IPM for solving second orderconic programming problems based on the work of Nesterov and Todd [28]. Luo [23],Sturm [34] and Zhang [40] extended the homogeneous IPM to general conic program-ming, including semidefinite programming. Recently, Skajaa et al. [33] implementeda homogeneous IPM for solving nonsymmetric conic programming problems.

SOLVING CONVEX PROGRAMMING USING INTERIOR POINT METHOD 3

Andersen and Ye [11, 12] extended the homogeneous IPM to the general MCP.They developed a homogeneous IPM that converges to a maximal complementarysolution in at most O(

√n log 1/ε) iterations if f satisfies SLC. Although the global

linear polynomial time convergence is provided in their analysis, exact search direc-tions and restrictive step size computations are required, and hence their algorithmis not practical.

Motivated by the apparent gap between the theoretical complexity results andlong-step practical implementations in IPM, Mehrotra and Huang [26] recently in-troduced a homogeneous IPM equipped with a continuously differentiable potentialfunction for LP. Their method ensures a global linear polynomial time convergencewhile providing the flexibility to integrate heuristics for generating the search direc-tions and the step size computations. More importantly, they showed that it is possibleto implement a potential reduction IPM that uses similar number of iterations as analgorithm that ignores global linear polynomial time convergence.

In this paper the Mehrotra-Huang potential function is extended to the con-text of MCP. Our goal is to develop a stable interior point solver that ensures theglobal linear polynomial time convergence without compromising the computationalefficiency. Two guiding principles are used in our implementation: (1) ensure a reduc-tion in the potential function; (2) use the Mehrotra “corrector” direction to improvethe reduction in the potential function. Our algorithm is implemented in a softwarepackage named iOptimize. iOptimize is developed based on a variant of Mehrotra’spredictor-corrector framework [25]. Computational results on the standard test prob-lems [2, 5, 6, 24] show that iOptimize appears to perform more robustly and solvesproblems with fewer average iterations than a mature commercial solver MOSEK (bar-rier solver version 6.0.0.106; MOSEK for short) [7]. In particular, all problems in testsets [2, 5, 6, 24] were solved by iOptimize using default settings, while MOSEK failed onseveral problems. Moreover, the average number of iterations used by iOptimize arefewer than those taken by MOSEK for convex quadratic programming (QP) problems,convex quadratically constrained quadratic programming (QCQP) problems, and gen-eral convex programming (CP) problems in the test sets. We further created someinfeasible problems to check the robustness of our approach. While iOptimize solvedall these problems (except one, where the failure was due to an AMPL error message),MOSEK failed on several problems. We also found that iOptimize seems to detect in-feasibility more reliably than general nonlinear solvers Ipopt (version 3.9.2) [3] andKnitro (version 8.0) [4].

The rest of this paper is organized as follows. In Section 2 we outline the knownresults for the homogeneous formulation for MCP. In Section 3 we describe our poten-tial function and show that our potential reduction algorithm converges to a maximalcomplementary solution in polynomial time. In Section 4 we provide additional imple-mentation details of iOptimize. Computational results are presented and discussedin Section 5. Concluding remarks are made in Section 6.

2. Homogeneous formulation for the monotone complementarity prob-lem. In this section we summarize the main results of Andersen and Ye’s [11, 12,Theorem 1] homogeneous IPM for MCP. Andersen and Ye consider an augmentedhomogeneous model related to MCP (HMCP):

s = τf(x/τ),(2.1)

κ = −xT f(x/τ),(2.2)


0 ≤ (s, κ) ⊥ (x, τ) ≥ 0.(2.3)

The right-hand side in (2.1) is closely related to the recession function in convexanalysis of Rockafellar [32, Section 8]. Given a maximal complementary solution(x∗, τ∗, s∗, κ∗) that satisfies (2.1) and (2.2), suppose τ∗ > 0 and hence κ∗ = 0. Thenfrom (2.1) and (2.2) we have

s∗/τ∗ = f(x∗/τ∗) and κ∗/τ∗ = −(x∗/τ∗)T (s∗/τ∗) = 0.

This implies that (x∗/τ∗, s∗/τ∗) is a complementary solution for MCP. On the otherhand, if κ∗ > 0 and hence τ∗ = 0, then the complementarity of MCP at (x∗, s∗) is notzero. In this case MCP is said to be infeasible, and x∗ is a direction of a separatinghyperplane that separates 0 and the set s− f(x) ∈ Rn | (x, s) > 0 .

Let

F (x, τ) :=

(τf(x/τ)−xT f(x/τ)

): Rn+1

++ → Rn+1.

Andersen and Ye [12] showed the following theorem.Theorem 2.1. Consider the MCP and the associated HMCP.1. If ∇f is positive semidefinite in Rn+, then ∇F is also positive semidefinite in

Rn+1++ .

2. F is a continuous homogeneous function in Rn+1++ with degree 1 and for any

(x, τ) ∈ Rn+1++ , (x; τ)TF (x, τ) = 0, and (x, τ)T∇F (x, τ) = −F (x, τ)T .

3. If f is a continuous monotone mapping from Rn+ to Rn, then F is a contin-

uous monotone mapping from Rn+1++ to Rn+1.

4. If f is scaled Lipschitz with νf , then F is also scaled Lipschitz; that is, itsatisfies condition (1.6) with

νF (β) =

(1 +

2νf (2β/(1− β))

1− β

)(1

1− β

),

where β < 1/3.5. There exists a bounded sequence 〈xk, τk, sk, κk〉 ⊂ R2n+2

++ , k = 0, 1, . . . thatsolves HMCP in a limit.

6. Let (x∗, τ∗, s∗, κ∗) be a maximal complementary solution or ray for HMCP.MCP has a solution if and only if τ∗ > 0. In this case, (x∗/τ∗, s∗/τ∗) is acomplementary solution for MCP.

7. MCP is (strongly) infeasible if and only if κ∗ > 0. In this case, (x∗/κ∗, s∗/κ∗)is a certificate (ray) to prove (strong) infeasibility.

We note that if f is an affine function, then there is a strictly complementarysolution for HMCP. On the other hand, if f is a monotone function, then the maximalcomplementary solution (x∗, τ∗, s∗, κ∗) may not be a strictly complementary solution.However, in this case Andersen and Ye [12, proof for Theorem 1] showed that τ∗+κ∗ >0. This implies (x∗, s∗) must either generate a finite solution for HMCP, or give acertificate that proves the infeasibility of MCP.

2.1. The path following homogeneous algorithm. For simplicity in thefollowing we let n := n+ 1, x := (x; τ) ∈ Rn+ and s := (s;κ) ∈ Rn+. Let

rk := sk − F (xk),


and let (x0, s0) = e be the starting point. Let C(ϑ) denote a continuous trajectorysuch that

C(ϑ) :=

(xk, sk) | sk − F (xk) = ϑr0, Xksk = ϑe, 0 < ϑ ≤ 1.

Note that such a trajectory always exists (Guler [19]). Moreover, Andersen and Ye [12,Theorem 2] showed that this continuous trajectory is bounded and any limit point is amaximal complementary solution for HMCP. At iteration k with iterate (xk, sk) > 0,the algorithm computes the search direction (dx, ds) by solving the following systemof linear equations:

ds −∇F (xk)dx = −ηrk,(2.4)

Xkds + Skdx = γµke−Xksk.(2.5)

Here η and γ are parameters between 0 and 1, and µk := (xk)T sk

n . For a step sizeα > 0, let the new iterate be

x+ := xk + αdx > 0 and

s+ := sk + αds + (F (x+)− F (xk)− α∇F (xk)dx)) = F (x+) + (1− αη)rk > 0.(2.6)

Andersen and Ye [12] showed the following lemmas.Lemma 2.2. The direction (dx, ds) satisfies

dTx ds = dTx∇F (xk)dx + η(1− γ − η)nµk.

Lemma 2.3. Let r+ = s+ − F (x+), and consider the new iterate (x+, s+) givenby (2.4)–(2.6). Then,

1. r+ = (1− αη)rk.2. (x+)T s+ = (xk)T sk(1− α(1− γ)) + α2η(1− η − γ)nµk.

Note that from Lemma 2.3, for η = 1 − γ the infeasibility residual and thecomplementarity are reduced at the same rate. Based on these lemmas, Andersenand Ye [12] showed that with suitable chosen parameters α and β, their homogeneousalgorithm will generate a sequence 〈xk, sk〉 > 0 such that 〈xk, sk〉 ≤ ‖XKsk −µke‖ ≤ϑµk, where ϑ ∈ (0, 1] is a constant parameter related to νF (β). Moreover, theyshowed that both ‖rk‖ and (xk)T sk converge to zero at a global rate γ = 1− ϑ/

√n.

As a result, if f satisfies the SLC, then the homogeneous algorithm converges inO(√n log(1/ε)) iterations. However, it requires exact search directions and uses short

step computations, and thus is not a practical algorithm.

3. A practical potential reduction homogeneous algorithm. We considerthe following potential function for the HMCP.

Φρ(x, s) := (ρ/2) log

(xT s)2 + θ‖r‖2−

n∑j=1

log xjsj ,(3.1)

where θ is a constant parameter with positive value and ρ ≥ n +√n. A potential

function of this form was first introduced by Goldfarb and Mehrotra [17], and wasextended to the homogeneous LP model by Mehrotra and Huang [26]. It is similar to


(1.7) except that the residual vector r is considered in the function. Without loss ofgenerality we choose θ = 1 in (3.1).

We need to analyze the worst case decrease in (3.1) at each iteration of ourpotential reduction IPM. Let

Sk := diag(sk1 , . . . , skn), D := (XkSk)0.5, Dmin := minj=1,...,n

(xkj s

kj

)0.5,

p := nρµe−D

2e = xkTsk

ρ e−D2e, and

α := β Dmin

‖D−1p‖ ,

(3.2)

where β ∈ (0, 1) is a positive constant that will be determined later. Suppose (dx, ds)is computed by solving the linear system of equations (2.4) and (2.5) with η = 1− γand γ = n

ρ . Let

px := αdx,

ps := αds + (F (x+)− F (xk)− α∇F (xk)dx),(3.3)

z := Xk(F (x+)− F (xk)−∇F (xk)px), and

q := z/α,

and let x+ := xk + px > 0, and s+ := sk + ps > 0. We have

p = γµe−D2e = Xkds + Skdx,(3.4)

and

Xkps + Skpx − αp = z = αq.(3.5)

In Section 3.1 we show that if f satisfies SLC (and consequently so does F ), thenat iteration k the theoretical direction (dx, ds) with an appropriate step size α resultsin

Φρ(x+, s+)− Φρ(x

k, sk) ≤ ζ,

where ζ is a negative constant. Let (x+, s+) be an iterate computed by using anyfavored search direction with inexact step sizes in the primal and dual spaces. Start-ing from an initial solution (x0, s0) > 0, our potential reduction IPM is given inAlgorithm 1. This algorithm is designed to have the freedom in choosing any favoredsearch direction with inexact and different step sizes in the primal and dual spaces.

In Section 3.2 we show that if Φρ(xk, sk) is reduced by at least a constant amount

at each iteration and Φn(xk, sk) (potential function 3.1 with parameter ρ = n) isupper bounded by a constant amount ιn log n (ι > 0 is a constant that will be givenlater), and iterates are maintained while satisfying Assumption 3.1 given below, thenAlgorithm 1 generates a maximal complementary solution of desired precision in poly-nomial time.

Assumption 3.1.

(i) Ω(1) ≤ (x0)T s0|rkj |(xk)T sk|r0j |

≤ O(1), j = 1, . . . , n.

(ii) Ω(1) ≤ (x+)T s+

(x+)T s+≤ O(1).

Assumption 3.1 (i) requires that the overall decrease in the infeasibility to com-plementarity ratio is maintained in Algorithm 1. Assumption 3.1 (ii) requires thatthe complementarity at the iterate (x+, s+) generated using a favored direction and


Algorithm 1 Practical potential reduction IPM

Input: An initial solution (x0, s0) > 0, constants ζ < 0 and ι > 0.Output: An (asymptotically) maximal complementary solution (xk, sk).

1: Let k = 0.2: while Termination criteria is not met do3: Compute (x+, s+) using any favored search direction and step size for xk and

sk.4: if Φρ(x

+, s+)− Φρ(xk, sk) ≤ ζ, Φn(xk, sk) ≤ ιn log n, and if Assumption 3.1 is

satisfied then5: Set (xk+1, sk+1) := (x+, s+).6: else7: Compute (x+, s+) using the theoretical direction (dx, ds) with an appropriate

step size α.8: Set (xk+1, sk+1) := (x+, s+).9: end if

10: Set k := k + 1.11: end while

the theoretical iterate (x+, s+) are of the same order. Note that if η = 1 − γ, thenfrom Lemma 2.3 we have (x+)T s+ = (1− αη)(xk)T sk. As a result if only theoreticaldirection is chosen in the algorithm, Assumption 3.1 is satisfied with constant equalto 1, and the complementarity is decreased monotonically. Convergence propertiesfor an algorithm that has additional flexibility of choosing more aggressive directionsand step size are still ensured under Assumption 3.1.

3.1. Potential function reduction using the theoretical direction. Thefollowing analysis follows some of the steps in Kojima et al. [22] and Potra and Ye [31,Sections 2 and 3]. We drop the index k for an iteration to simplify the presentation.

Proposition 3.2 (Kojuma et al. [22, Lemma 2.3]). Let u, v,and w ∈ Rn satisfyu+ v = w and uT v ≥ 0. Then

‖u‖ ≤ ‖w‖, ‖v‖ ≤ ‖w‖, and uT v ≤ 1

2‖w‖2.

Lemma 3.3. Let px be defined as in (3.3). Then, ‖X−1px‖∞ ≤ β.Proof. We use the technique in Kojima et al. [22] for the proof. We first rewrite

equation (3.4) as

D−1Xds‖D−1p‖

+D−1Sdx‖D−1p‖

=D−1p

‖D−1p‖.(3.6)

From Lemma 2.2, we have

(D−1Xds)TD−1Sdx

‖D−1p‖‖D−1p‖=

dTx ds‖D−1p‖2

≥ 0.(3.7)

Now using Proposition 3.2 in (3.6) and (3.7), we have

‖D−1Sdx‖‖D−1p‖

≤ ‖D−1p‖

‖D−1p‖= 1.(3.8)


Using (3.3) and (3.8), we deduce the following relation

‖X−1dx‖‖D−1p‖

=‖D−2Sdx‖‖D−1p‖

≤ D−1min.(3.9)

Finally, using (3.2) and (3.9), we have

‖X−1px‖∞ = α‖X−1dx‖∞ ≤ βDmin‖X−1dx‖‖D−1p‖

≤ β.

Potra and Ye [31] showed the following lemma.Lemma 3.4 (Potra and Ye [31, Lemma 3.2]). If dx and ds are the solution of the

linear system (3.5), and if condition (1.6) and Lemma (3.3) are satisfied, then

‖z‖ ≤ ξβ2D2min,

where ξ = 0.25νF (β) and z is defined in (3.3).From the above lemma and (3.2) we have

‖D−1q‖ ≤ ‖q‖D−1min =

‖z‖αD−1

min ≤ξβ2

αDmin = ξβ‖D−1p‖.(3.10)

Kojima et al. showed the following lemma.Lemma 3.5 (Kojima et al. [22, Lemma 2.5]). Suppose the search direction (dx, ds)

is computed by solving the linear system of equations (2.4) and (2.5) with η = 1 − γand γ = n

ρ (ρ ≥ n+√n), then

ρ

xT s‖D−1p‖ =

∥∥∥D−1e− ρ

xT sDe∥∥∥ ≥ √3

2D−1

min.

In the following we define two real valued functions g1 and g2, and establish resultsthat upper bound them. These two functions and the related upper bounds will beused later in our analysis. Let

g1 := −αρη + eT(X−1px + S−1ps

)and(3.11)

g2 :=‖X−1px‖2 + ‖S−1ps‖2

2(1− β).(3.12)

Lemma 3.6. Let g1 be defined as in (3.11). With a suitably chosen parameterβ ∈ (0, 1), suppose the search direction (dx, ds) is computed by solving the linearsystem of equations (2.4) and (2.5) with η = 1− γ and γ = n

ρ (ρ ≥ n+√n). Let the

step size α be defined as in (3.2). If condition (1.6) is satisfied, then

g1 ≤ −√

3

2(1− ξβ)β + ξβ2 ρ√

n.

Proof. From (2.5) and γ = 1− η, we have

−αρη = αρxT ds + sT dx

nµ.(3.13)


An upper bound of g1 is obtained in the following set of inequalities.

g1 = αρ

(xT ds + sT dx

xT s

)− eT

(X−1px + S−1ps

)= α

ρ

xT seT (Xds + Sdx)− eTD−2(Xps + Spx)

= αρ

xT seT(p− xT s

ρD−2(p+ q)

)(using (3.4) and (3.5))

= αρ

xT seT((

D − xT s

ρD−1

)D−1p− xT s

ρD−2q

)= −α ρ

xT s

(pTD−1D−1p

)− α ρ

xT s

(D−1p+De

)TD−1q (using (3.2))

= −α ρ

xT s

(pTD−1D−1(p+ q)

)− α ρ

xT seT q

= −α ρ

xT s

(‖D−1p‖2 + pTD−1D−1q

)− ρ

xT seT z (using (3.3))

≤ −α ρ

xT s

(‖D−1p‖2 − ‖D−1p‖‖D−1q‖

)+

ρ

xT s‖z‖1

≤ −α ρ

xT s

(‖D−1p‖2 − ξβ‖D−1p‖2

)+

ρ

xT s

√n‖z‖ (using (3.10))

≤ −α ρ

xT s(1− ξβ)‖D−1p‖2 +

ρ

xT s

√nξβ2D2

min (using Lemma 3.4)

≤ −√

3

2(1− ξβ)αD−1

min‖D−1p‖+

ρ√nξβ2 (using Lemma 3.5)

= −√

3

2(1− ξβ)β +

ρ√nξβ2. (using (3.2))

Lemma 3.7. Let g2 be defined as in (3.12). Under the hypothesis of Lemma 3.7,we have

g2 ≤β2(1 + ξβ)2

2(1− β).

Proof. The following proof is based on techniques in Potra and Ye [31]. We have

2(1− β)g2 = ‖X−1px‖2 + ‖S−1ps‖2(3.14)

= ‖D−2Xps‖2 + ‖D−2Spx‖2

≤ D−2min

(‖D−1Xps‖2 + ‖D−1Spx‖2

)= D−2

min

(‖D−1(Xps + Spx)‖2 − 2pTx ps

)= D−2

min

(α2‖D−1(p+ q)‖2 − 2pTx ps

)(using (3.5))

≤ D−2min

(α2(‖D−1p‖+ ‖D−1q‖)2 − 2pTx ps

)≤ D−2

min

(α2 (1 + ξβ)2|D−1p‖2 − 2pTx ps

)(using (3.10))

=(αD−1

min‖D−1p‖

)2(1 + ξβ)2 − 2D−2

minpTx ps

= β2(1 + ξβ)2 − 2D−2minp

Tx ps. (using (3.2))

The upper bound on pTx ps is obtained as follows.

pTx ps = pTx (αds + (F (x+)− F (x)−∇F (x)px)) (using (3.3))(3.15)


= (x+ − x)T (F (x+)− F (x)) + α2(dTx ds − dTx∇F (x)dx) (using (3.3))

= (x+ − x)T (F (x+)− F (x)) + α2η(1− η − γ)nµ (using Lemma 2.2)

≥ 0 (from the monotonicity of F ).

The result follows from using (3.14) and (3.15).The following proposition is frequently used in the IPM analysis.

Proposition 3.8 (Karmarker [21]).1. If 1− δ > 0, then log 1− δ ≤ −δ.2. If δ ∈ Rn satisfies ‖δ‖∞ ≤ β < 1, then

∑nj=1 log 1− δj ≥ −eT δ − ‖δ‖2

2(1−β) .

Now we are ready to show a bound on the difference between the potential functionvalues Φρ(x

+, s+) and Φρ(x, s). The next theorem bounds the improvement in thepotential function from taking the theoretical direction with a controllable step size.

Theorem 3.9. Let ξ = 0.25νF (β) ≥ 0.25. Suppose the search direction (dx, ds)is computed by solving the linear system of equations (2.4) and (2.5) with η = 1 − γand γ = n

ρ (ρ ≥ n+√n). Let the step size α be defined as in (3.2). If condition (1.6)

is satisfied, and β ∈ (0, 0.303) is chosen such that

ξβ ≤√

3

4(√

3 + 2ρ/√n), and(3.16)

β√ξ(1 + ξβ) ≤

√3

2(√

3 + 2ρ/√n)1/2 .(3.17)

Then, (x+, s+) is always strictly positive. Moreover,

Φρ(x+, s+)− Φρ(x, s) ≤ −0.278

β

1− β.

Proof. We first show that (x+, s+) is strictly positive. From Lemma 3.7 and(3.17), we have

max‖X−1px‖∞, ‖S−1ps‖∞

≤ β(1 + ξβ) ≤

√3

2(ξ(√

3 + 2ρ/√n))1/2 < 1.

Hence, (px, ps) is a valid direction. We note that from Lemma 3.3, we have

‖X−1px‖∞ ≤ β ≤√

3

4ξ(√

3 + 2ρ/√n)≤

√3

2(ξ(√

3 + 2ρ/√n))1/2 .

Therefore, the upper bound from Lemma 3.3 is still valid (and tighter). Moreover,β < 0.303 implies that the condition in Theorem 2.1 [4] is satisfied. Now,

Φρ(x+, s+)− Φρ(x, s)

= (ρ/2) log

(1− αη)2((nµ)2 + ‖r‖2

)−

n∑j=1

log (xj + (px)j) (sj + (ps)j)

− (ρ/2) log

(nµ)2 + ‖r‖2

+

n∑j=1

log xjsj (using Lemma 2.3)


= ρ log 1− αη −n∑j=1

log

1 +

(px)jxj

1 +

(ps)jsj

≤ −αρη − eT(X−1px + S−1ps

)− ‖X

−1px‖2 + ‖S−1ps‖2

2(1− β)(using Proposition 3.8)

= g1 + g2.

From Lemmas 3.6 and 3.7, we write

Φρ(x+, s+)− Φρ(x, s) ≤ −

√3

2(1− ξβ)β + ξβ2ρ/

√n+

β2(1 + ξβ)2

2(1− β)

=β2(1 + ξβ)2

2(1− β)+ β

(−√

3

2+ ξβ

(√3

2+

ρ√n

))

=β2(1 + ξβ)2

2(1− β)−√

3

2β

(1− ξβ

(√3 + 2(ρ/

√n)√

3

))=: ζ.

It follows that

−21− ββ

ζ = −β(1 + ξβ)2 +√

3(1− β)

(1− ξβ

(√3 + 2(ρ/

√n)√

3

)).

Numerically, it is easy to verify that

ζ ≤ −0.278β

1− β.

Since νF (β) : (0, 1) 7→ (1,∞) is a monotone increasing function depending onβ ∈ (0, 0.303), we can always choose a sufficiently small β that satisfies (3.16) and(3.17). As a result, a reduction in the potential function (ζ < −0.278 β

1−β ) is guaran-

teed. However, choosing β is dependent on νF (β), finding a maximal β that satisfies(3.16) and (3.17) requires the knowledge of the scaled Lipschitz constant νF (β), whichmay not be available in practice for an arbitrary convex function. Problems satisfyingthe scaled Lipschitz condition are discussed in [27, 30, 41]. Nevertheless, the analysissuggests that for an appropriate step size, the potential function value will reduce aslong as the scaled Lipschitz constant is bounded. Consequently, the potential functioncan be used as a guide to solve monotone complementarity and convex optimizationproblems satisfying this property. If the condition is not satisfied, then lack of suffi-cient reduction in the potential function indicates that the algorithm proposed hereis not appropriate for the problem being solved. Finally, we note that Theorem 3.9 issimilar to the result by Potra and Ye [31], but with a difference that it allows us tochoose a larger ρ, when β is sufficiently small.

Remark. Note that by performing simple block elimination on the system of linearequations (2.4) and (2.5), we obtain the following alternative formulation:

X(α∇F (x)dx − αηr) + αSdx − αp = 0.(3.18)

Hence, dx is a direction obtained by applying a single Newton step to the equation

z(h) := X(F (x+ αh)− F (x)− αηr) + αSh− αp = 0,(3.19)


with starting point h = 0. If F is linear (for example, LP formulated as a HMCPmodel), then the linear system (3.19) can be solved exactly. In this case Theorem 3.9with ξ = 0 reduces to the result of Mehrotra and Huang [26].

Suppose F is nonlinear and does not satisfy SLC. Given positive constants ξ and βthat satisfy conditions (3.16) and (3.17). If one can compute an approximate solutionδx such that

‖z(δx)‖ = ‖X(F (x+ αδx)− F (x)− αηr) + αSδx − αp‖ ≤ ξβ2 minj=1,...,n

(xjsj),(3.20)

then the result of Theorem 3.9 is still satisfied. However, finding a direction δx mayrequire many Newton steps, and an explicit bound on the number of Newton steps isunknown.

3.2. Convergence of the practical potential reduction homogeneous al-gorithm. The following theorem shows that if potential function (3.1) can be reducedby at least a constant amount at each iteration, then the Algorithm 1 generates anε−complementary solution in polynomial time. Our analysis follows some of the stepsin Kojima et al. [22] and Porta and Ye [29].

Theorem 3.10. Let ε > 0. Starting from a solution (x0, s0) > 0, if the value ofpotential function (3.1) can be reduced by at least a constant amount and if Assumption3.1 is satisfied at each iteration of Algorithm 1, then after at most O(Φρ(x

0, s0) −n log n+ (ρ− n) |log ε|) iterations, we will have a solution such that

xT s ≤ ε and|rj |∣∣r0j

∣∣ ≤ O(1)xT s

(x0)T s0, j = 1, . . . , n.(3.21)

Proof. Using Assumption 3.1, we have

(ρ/2) log

(xT s)2

1 +

(Ω(1)

∥∥r0∥∥

(x0)T s0

)2−

n∑j=1

log xjsj ≤ Φρ(x, s),

which implies

(ρ− n) log xT s−n∑j=1

log nxjsjxT s

+ (ρ/2) log

1 +

(Ω(1)

∥∥r0∥∥

(x0)T s0

)2(3.22)

≤ Φρ(x, s)− n log n.

Since the second and the third terms of (3.22) are always nonnegative, it follows that

(ρ− n) log xT s ≤ Φρ(x, s)− n log n.

Therefore, Φρ(x, s) ≤ n log n− (ρ− n) |log ε| , then we have

xT s ≤ ε.

Moreover, Assumption 3.1 implies that

|rj |∣∣r0j

∣∣ ≤ O(1)xT s

(x0)T s0, j = 1, . . . , n.


We now show that any limit point of a sequence 〈xk, sk〉 generated by the potentialreduction algorithm of Section 3.2 is a maximal complementary solution. Our analysishere follows some of the steps in Guler and Ye [20, Sections 2 and 4] and Potra andYe [31, Section 4].

Lemma 3.11. Given an initial positive point (x0, s0). Recall that (x+, s+) isa tentative point computed using any favored search direction using an inexact anddifferent step size in the primal and dual spaces, and it satisfies Assumption 3.1;whereas (x+, s+) is computed using the theoretical direction (dx, ds) and a fixed stepsize α. Let

C1 :=ρ− n

2log

((x0)T s0

)2+O(1)2

∥∥r0∥∥2

((x0)T s0)2

+ Ω(1)2 ‖r0‖2

≥ 0, and

C2 := C1 + max 0,−(ρ− n) log Ω(1) ≥ 0,

then,

Φn(x+, s+)− Φn(xk, sk) ≤ ζ − (ρ− n) log

1−D2

min

√n

(xk)T sk

, and(3.23)

Φn(x+, s+)− Φn(xk, sk) ≤ ζ − (ρ− n) log

1−D2

min

√n

(xk)T sk

+ C2,(3.24)

where Φn(xk, sk) is defined as in (3.1) with ρ = n.Proof. To simplify the presentation we drop the index k. Using Assumption 3.1,

we have

∥∥r+∥∥2 ≥

(Ω(1)

∥∥r0∥∥ (x+)T s+

(x0)T s0

)2

, ‖r‖2 ≤(O(1)

∥∥r0∥∥ xT s

(x0)T s0

)2

, and(3.25)

(x+)T s+ ≥ Ω(1)(x+)T s+.(3.26)

First we bound the difference between the values of Φn(x+, s+) and Φn(x, s). Letr+ = s+ − F (x+).

Φn(x+, s+)− Φn(x, s)

= Φρ(x+, s+)− Φρ(x, s)− Φρ(x

+, s+) + Φρ(x, s) + Φn(x+, s+)− Φn(x, s)

≤ ζ − ρ− n2

log

((x+)T s+)2 +∥∥r+

∥∥2

+ρ− n

2log

(xT s)2 + ‖r‖2

(using Theorem 3.9)

≤ ζ − ρ− n2

log

((x+)T s+)2 +

(Ω(1)

∥∥r0∥∥ (x+)T s+

(x0)T s0

)2

+ρ− n

2log

(xT s)2 +

(O(1)

∥∥r0∥∥ xT s

(x0)T s0

)2

(using (3.25))

= ζ − (ρ− n) log

(x+)T s+

xT s

+ C1

≤ ζ − (ρ− n) log

(x+)T s+

xT s

− (ρ− n) log Ω(1) + C1 (using (3.26))


≤ ζ − (ρ− n) log

(x+)T s+

xT s

+ C2

≤ ζ − (ρ− n) log 1− αη+ C2 (using Lemma 2.3)

= ζ − (ρ− n) log

1 + α

xT ds + sT dsxT s

+ C2 (using (3.13))

= ζ − (ρ− n) log

1 + α

pT e

xT s

+ C2 (using (3.4)).

Now,

αpT e

xT s=α((xT s/ρ

)eT e− eTD2e

)xT s

(using (3.2))

= α

(n

ρ− 1

)= −β Dmin

‖D−1p‖

(ρ− nρ

)(using (3.2))

≥ −βD2min

2(ρ− n)√3xT s

(using Lemma 3.5)

≥ −D2min

√n(ρ− n)

2xT sξ(√

3n+ 2ρ)(using (3.16))

≥ −D2min

2ρ√n

xT s(√

3n+ 2ρ)(because ξ = 0.25νFβ ≥ 0.25)

≥ −D2min

√n

xT s.

Therefore,

Φn(x+, s+)− Φn(x, s) ≤ ζ − (ρ− n) log

1−D2

min

√n

xT s

+ C2.

The difference between the values of Φn(x+, s+) and Φn(x, s) can be constructed ina similar way.

Lemma 3.12. Given an initial positive point (x0, s0). Suppose a sequence 〈xk, sk〉is generated by using Algorithm 1. There is a constant M1 independent of k such that

Φn(xk, sk) ≤M1 for all k.

Proof. Let C3 := (n/2) log

1 +

(O(1)

‖r0‖(x0)T s0

)2> 0. Using (3.25), we have

Φn(x, s) = (n/2) log

(xT s)2 + ‖r‖2−

n∑j=1

log xjsj

≤ (n/2) log

(xT s)2 +

(O(1)

∥∥r0∥∥ xT s

(x0)T s0

)2−

n∑j=1

log xjsj (using (3.25))

= n log xT s−n∑j=1

log xjsj + C3.(3.27)


In the following we analyze the behavior of n log xT s −∑nj=1 log xjsj . Let ι > 0 be

sufficiently large so that

ζ − (ρ− n) log

1− 1

nι−0.5

< 0.(3.28)

Now consider the case where Φn(x, s) ≥ n log xT s−∑nj=1 log xjsj ≥ ιn log n. In this

case we have,

n log xT s+ n logD−2min ≥ ιn log n,

⇒ logxT sD−2

min

≥ ι log n,

⇒ Φn(x+, s+)− Φn(x, s) = ζ − (ρ− n) log

1−D2

min

√n

xT s

(using (3.23))

≤ ζ − (ρ− n) log

1− 1

nι−0.5

< 0.

The above relations suggests that using the theoretical direction and a fixed step sizecan decrease the value of Φn(x, s) if this value is larger than ιn log n. Next we considerthe case where Φn(x, s) ≤ ιn log n. We show that the value of Φn(x, s) in this case isupper bounded by a constant amount independent of k, and the iterate is computedusing any favored search direction with inexact and different step sizes in the primaland dual spaces.Using (3.24) and (3.27), we have

Φn(x+, s+) ≤ Φn(x, s) ≤ ιn log n+ C3

≤ ζ − (ρ− n) log

1−D2

min

√n

xT s

+ ιn log n+ C2 + C3

≤ ζ − (ρ− n) log

1− 1√

n

+ ιn log n+ C2 + C3.

Consequently, it follows that Lemma 3.12 holds with

M1 := max

ζ − (ρ− n) log

1− 1√

n

+ ιn log n+ C2 + C3,Φn(x0, s0)

.

Lemma 3.13. Let M1 be a fixed constant, and

Φn(xk, sk) ≤M1 for all k.

Under the hypothesis of Lemma 3.12. Then, there exists a fixed M2 (independent ofk) such that

minxkj s

kj

(xk)T sk

≥M2.

Proof. As usual, the index k is dropped for simplification. Using the result inLemma 3.12, we have

M1 ≥ Φn(x, s) = (n/2) log

(xT s)2 + ‖r‖2−

n∑j=1

log xjsj ≥ n log xT s−n∑j=1

log xjsj .


Hence, it follows that

log xT s− log xisi ≤M1 − (n− 1) log xT s+∑j 6=i

log xjsj

≤M1 − (n− 1) logxT s− xisi

+∑j 6=i

log xjsj

≤M1 − (n− 1) log n− 1 =: − logM2.

Now let

σ(x) := j ∈ 1, 2, . . . , n : xj ≥ sj and

σ(s) := j ∈ 1, 2, . . . , n : sj ≥ xj .

Combining Lemmas 3.12 and 3.13, the following theorem shows that the sequence〈xk, sk〉 generates a maximal complementary solution for HMCP in the limit.

Theorem 3.14. Under the hypothesis of Lemma 3.12, the limit point of thesequence 〈xk, sk〉 is a maximal complementary solution for HMCP.

Proof. Let (x∗, s∗) be any maximal complementary solution for HMCP such that

s∗ = F (x∗) and (x∗)T s∗ = 0.

Let (x∞, s∞) be a limit point of the sequence 〈xk, sk〉. In the following we showthat (x∞, s∞) is a maximal complementary solution, that is, σ(x∞) = σ(x∗) and

σ(s∞) = σ(s∗). Let |r|T := (|r1| , . . . , |rn|) and C4 := O(1)|r0|T x∗

(x0)T s0≥ 0. Using

Assumption 3.1, we have

rTx∗ ≤ |r|T x∗

≤ O(1)(xT s)

∣∣r0∣∣T x∗

(x0)T s0

= C4xT s.(3.29)

From the monotonicity of F (x), for any positive solution (x, s) > 0, we have

(x− x∗)T (F (x)− F (x∗)) ≥ 0,

⇒ −xT s∗ − (s− r)Tx∗ ≥ 0,

⇒ −xT s∗ + sTx∗ ≤ rTx∗ ≤ C4xT s, (using (3.29))

⇒∑

j∈σ(x∗)

x∗jsj +∑

j∈σ(s∗)

s∗jxj ≤ C4xT s.

As a result, if j ∈ σ(x∗), then

C4xT s ≥ sjx∗j ≥ (sjxj)

x∗jxj≥ min xjsj

x∗jxj.

Using Lemma 3.13, the above relation implies

xj ≥min xjsjC4xT s

x∗j ≥M2

C4x∗j .

Hence, we have x∞j ≥ (M2/C4)x∗j > 0, consequently σ(x∗) = σ(x∞). The proof ofσ(s∗) = σ(s∞) can be constructed in a similar way.


4. Implementation details. In this section we present the implementation de-tails of a potential reduction homogeneous algorithm for the MCP using the potentialfunction given in Section 3. We describe the details in the context of solving a generalconvex optimization problem (CP).

4.1. Homogeneous formulation for convex programming. Consider a pri-mal CP

min c(x)(4.1)

s.t. ai(x) ≥ 0, i = 1, . . . , m,

ai(x) = 0, i = m+ 1, . . . ,m,

x ≥ 0, and x is free,

where x ∈ Rn represents the primal variables. We let n + n = n and let x =(x; x) ∈ Rn × Rn, where x and x respectively represent the “normal” and “free”primal variables. Here, the function c(·) : Rn → R is convex, the component functionsai(·) : Rn → R, i = 1, . . . , m are concave, and the component functions ai(·) : Rn →R, i = m+ 1, . . . ,m are affine.

We let m = m + m and y = (y; y) ∈ Rm × Rm, where y and y respectivelyrepresent the “normal” and “free” dual variables. Let s ∈ Rn represent the vector ofdual slacks. Let ω := (x; y; τ ; z; s;κ). In the following we use the notation

Ωx :=x = (x; x) : x ∈ Rn++, x ∈ Rn

,

Ωy :=y = (y; y) : y ∈ Rm++, y ∈ Rm

,

Ω :=ω : ω ∈ Ωx × Ωy ×Rm+n+2

++

.

Moreover, we assume that functions c(·) and ai(·), i = 1, . . . , m, are at least twicedifferentiable on the set Ωx, and the matrix

[Dz 00 0

]∇a(x)

∇a(x)T −∇2xL(x, y)−

[Dx 00 0

]

is nonsingular for any positive semidefinite matrices Dx ∈ Rn×n and Dz ∈ Rm×m.Here

L(x, y) := c(x)− yTa(x)

denotes the Lagrangian function, and we use the notation

∇xL(x, y) :=

(∂L(x, y)

∂x1, . . . ,

∂L(x, y)

∂xn

)to denote the gradient vector of L(x, y) associated with x, and use

∇2xL(x, y) :=

∂2L(x,y)∂x2

1· · · ∂2L(x,y)

∂x1∂xn

.... . .

...∂2L(x,y)∂xn∂x1

· · · ∂2L(x,y)∂x2

n


to denote the Hessian matrix of L(x, y) associated with x.A dual problem associated with (4.1) is

min L(x, y)− xT s = L(x, y)−∇xL(x, y)x(4.2)

s.t. ∇xL(x, y)T = (s; 0),

x, y, s ≥ 0, and x, y are free.

Using (4.1) and (4.2), one can construct optimality conditions to (4.1) as

∇xL(x, y)T = (s; 0),(4.3)

a(x) = (z; 0),

0 ≤ (s, y) ⊥ (x, z) ≥ 0,

x, y are free.

Here z ∈ Rm represents the vector of surplus variables corresponding to the concavefunctions ai(·), i = 1, . . . , m.

Problem (4.3) is a monotone complementarity problem with equality constraintsand free variables. Following Andersen and Ye [11], one can formulate a homogenizedmonotone complementarity problem for (4.3) as follows:

τ∇xL(x/τ, y/τ)T = (s; 0),(4.4)

τa(x/τ) = (z; 0),

−xT∇xL(x/τ, y/τ)T − yTa(x/τ) = κ,

0 ≤ (s, y, κ) ⊥ (x, z, τ) ≥ 0,

x, y are free,

where τ and κ are introduced homogeneous variables.Given a point ω ∈ Ω, let us define the residual vectors as

rvd := τg − (s; 0),

rvp := τa(x/τ)− (z; 0),

rvg := −κ− xT g − yTa(x/τ),

and

J := ∇a(x/τ),

g := ∇xL(x/τ, y/τ)T = ∇c(x/τ)T − JT (y/τ).

Let

(rp; rd; rg) := ηr := −η(rvp; rvd; rvg).

Let X := diag(x) and Y := diag(y). Starting from ω, the (perturbed-)Newtonmethod solves the following system of linear equations to compute direction dω :=(dx; dz; dτ ; dy; ds; dκ): H v2 −JT

vT1 Hg −vT3J v3 0

dxdτdy

− (ds; 0)

dκ(dz; 0)

=

rdrgrp

,(4.5)


and

Xds + Sdx = rxs := γµ(ω)e− Xs,Zdy + Y dz = rzy := γµ(ω)e− Zy,τdκ + κdτ = rτκ := γµ(ω)− τκ,

(4.6)

where

µ(ω) := ((x; z; τ)T (s; y;κ))/(n+ m+ 1),

H := ∇2xL(x/τ, y/τ),

v1 := −∇c(x/τ)T −H(x/τ),

v2 := ∇c(x/τ)T −H(x/τ),

v3 := a(x/τ)− J(x/τ),

Hg := (x/τ)TH(x/τ).

We can verify that H v2 −JTvT1 Hg −vT3J v3 0

is a positive semidefinite matrix. Hence, the complementarity problem (4.4) is mono-tone.

The coefficient matrix in the system of linear equations (4.5) and (4.6) in generalis sparse. By performing simple block elimination on the linear system, we obtain thefollowing alternative formulation, which is an augmented system form [14] with anadditional row and column: Dz J v3

−JT Dx v2

−vT3 vT1 Dg

dydxdτ

=

rp + (Y −1rzy; 0)

rd + (X−1rxs; 0)rg + τ−1rτκ

:=

rprdrg

,(4.7)

where

Dx := H +

[X−1S 0

0 0

], Dz :=

[ZY −1 0

0 0

], and Dg := Hg + τ−1κ.

Now define two vectors u := (v3;−v2) and v := (−v3; v1), and a matrix M :=[Dz JJT −Dx

]. Solving (4.7) is equivalent to solving the following two augmented

systems:

Mq = (rp; rd) and(4.8)

Mp = u.

Therefore, we have obatined the following components of the step direction:

dτ =rg − vT qDg − vT p

and

(dy; dx) = q − pdτ .

The remaining components ds, dz, dκ of the step direction can be easily recoverd.


M in (4.8) is a symmetric and indefinite matrix. To solve this system we can usethe Bunch-Parlett symmetric matrix factorization method [14] or the quasi-definitemethod [37]. The Bunch-Parlett method is known to avoid numerical instabilityassociated with free variables. In the current test results iOptimize uses the sparsesymmetric indefinite matrix factorization library MA57 [16].

4.2. Primal dual updates. Andersen and Ye [11] consider two types of up-dates: a linear update and a nonlinear update. Let ω = (x; y; τ ; z; s;κ) ∈ Ω be afeasible solution and dω = (dx; dy; dτ ; dz; ds; dκ) be a search direction. The linearupdate computes a new solution ω+ as the following:

ω+ := ω + αdω,

where α ≥ 0 is a given step size. The nonlinear update is a direct result of (2.6). Asshown in Lemma 2.3, with γ = 1− η the nonlinear update of the dual variables tendto reduce the infeasibilities and the complementarity gap at the same rate, which istheoretically desirable.

Andersen and Ye [11] propose a merit function to measure the progress in theiralgorithm. If the merit function is reduced to zero, then a complementary solutionis obtained. They show that for a suitably choice of γ, the (perturbed-) Newtondirection is a descent direction for the merit function.

In our implementation we use the potential function (3.1) to evaluate the progressin the algorithm. Given a search direction and a step size, we compute two trialsolutions using the linear and nonlinear updates. We accept the trial solution thatreduces the potential function more. We give a pseudo code for updating a solution,and discuss the step size computations in iOptimize in the next section.

4.3. Step size computations. To find a controllable step size, Andersen andYe employ a simple backtracking line search method equipped with safeguards. Theyrequire their merit function to be reduced by checking the Goldstein-Armijo rule.

To avoid line search computations, some standard implementations have useda certain fixed distance (step factor) to the boundary. However, a fixed factor >0.99 may be overly aggressive in the earlier phase of IPM. Mehrotra [25] proposeda heuristic to compute the step factor adaptively. His method adaptively allows forlarger (and smaller) step factor values. In our implementation we employ this heuristicto compute step sizes in the primal and dual spaces, and use them to update the pointwith linear or nonlinear update. This heuristic does not guarantee the reduction ofthe potential function value (assuming ∇xL(x, y) satisfies a SLC). In such a case,we choose the theoretical descent direction from solving (4.5) and (4.6) with suitableparameters η and γ that guarantees the reduction of the potential function value, andemploy a golden-section line search to find a controllable step size (See Section 4.5 fora detailed strategy).

Procedure computeStepFactor gives a pseudo code for computing primal (σp) anddual (σd) step factors, Procedure computeStepSize for computing the step size (α) inthe primal and dual spaces, and Procedure updateSolution for updating a given pointω along a search direction dω using linear and nonlinear updates. In procedure com-puteStepFactor, two parameters, θl (lower bound) and θu (upper bound), are used tosafeguard against very tiny or overly aggressive step factors.

4.4. Predictor-corrector technique. The computation of a search directionfrom (4.5) and (4.6) involves a matrix factorization step and subsequent solves withthe factored matrix. The cost of the solves in general is much cheaper, and thus


Procedure 2 (σp, σd) =computeStepFactor(ω, dω)

Input: Current point ω = (x; y; τ ; z; s;κ) ∈ Ω, search direction dω =(dx; dy; dτ ; dz; ds; dκ) parameters θl and θu.

Output: Primal step factor σp and dual step factor σd.1: Find the blocking variables:

lx := arg minj=1,...,n

−xj/(dx)j if (dx)j < 0 ,

ls := arg minj=1,...,n

−sj/(ds)j if (ds)j < 0 ,

lz := arg mini=1,...,m

−zi/(dz)i if (dz)i < 0 ,

ly := arg mini=1,...,m

−yi/(dy)i if (dy)i < 0 ,

lτ := −τ/dτ if dτ < 0,

lκ := −κ/dκ if dκ < 0.

2: Compute an estimate of the duality gap:

µx :=

n∑j=1

(xj + (dx)j)(sj + (ds)j),

µz :=

m∑i=1

(zi + (dz)i)(yi + (dy)i),

µτ := (τ + dτ )(κ+ dκ),

µ := ((µx + µz + µτ )(1− θl)) / (n+ m+ 1) .

3: Compute the step factors for all the variables:

σx :=

(µ

slx + (ds)lx− xlx

)(1

(dx)lx

),

σs :=

(µ

xls + (dx)ls− sls

)(1

(ds)ls

),

σz :=

(µ

ylz + (dy)lz− zlz

)(1

(dz)lz

),

σy :=

(µ

zly + (dz)ly− yly

)(1

(dy)ly

),

στ :=

(µ

κlτ + (dκ)lτ− τlτ

)(1

(dτ )lτ

),

σκ :=

(µ

τlκ + (dτ )lκ− κlκ

)(1

(dκ)lκ

).

4: Compute primal and dual step factors:

σp := max θl,min σx, σz, στ , θu ,σd := max θl,min σs, σy, σκ, θu .


Procedure 3 α =computeStepSize(ω, dω)

Input: Current point ω = (x; z; τ ; y; s;κ) ∈ Ω and search direction dω =(dx; dz; dτ ; dy; ds; dκ).

Output: Primal and dual step size α.1: Compute the step sizes to the boundary

αx := minj=1,...,n

−xj/(dx)j if (dx)j < 0 ,

αs := minj=1,...,n

−sj/(ds)j if (ds)j < 0 ,

αz := mini=1,...,m

−zi/(dz)i if (dz)i < 0 ,

αy := mini=1,...,m

−yi/(dy)i if (dy)i < 0 ,

ατ := −τ/dτ if dτ < 0,

ακ := −κ/dκ if dκ < 0.

2: Call Procedure computeStepFactor(σp, σd, ω, dω) to compute the primal step fac-tor (σp) and the dual step factor (σd).

3: Compute primal and dual step sizes:

αp := σp ×min αx, αz, ατ ,αd := σd ×min αs, αy, ακ ,

4: Set α := min αp, αd.

Procedure 4 ω+ =updateSolution(ω, dω, α)

Input: Current point ω = (x; y; τ ; z; s;κ) ∈ Ω, search direction dω =(dx; dy; dτ ; dz; ds; dκ), primal and dual step size α.

Output: Updated point ω+ ∈ Ω.1: Compute the primal and dual updates:ωl := ω + α(dx; dy; dτ ; dz; ds; dκ).

2: Compute the nonlinear update:

xn := x+ αdx,

yn := y + αdx,

τn := τ + αdτ ,

zni := ((1− αη)(−rvp)i + τnai(xn/τn)) , i = 1, . . . , m,

snj :=((1− αη)(−rvd)j + τn∇xLj(xn/τn, yn/τn)T

), j = 1, . . . , n,

κn :=((1− αη)(−rvg)− (xn)T∇xL(xn/τn, yn/τn)T − (yn)Ta(xn/τn)

).

3: Check if it is beneficial to take linear update:4: if Φρ(ω

n) ≥ Φρ(ωl) then

5: Set ω+ := ωl.6: else7: Set ω+ := ωn.8: end if


it leads to the idea of reusing the factorization of the linear system to compute abetter search direction. Several approaches have been developed based on this idea.For example, Mehrotra corrector [25] and Gondzio’s higher-order corrector [18] areproper choices. The presented empirical results are based on using Mehrotra correctoronly. We now describe an adaption of this technique for our context.

Let ωk ∈ Ω be the current point, we first compute an affine (pure Newton)direction by solving (4.5) and (4.6) with η = 1 and γ = 0, that is,

rp := −rvku, rxs := −Xksk,rd := −rvkp , rzy := −Zkyk,rg := −rvkd , rτκ := −τkκk.

The resulting direction is denoted by daω = (dax; daz ; daτ ; day; das ; daκ). Next we compute a

step size αa by calling Procedure αa =computeStepSize(ωk, daω) where ωk and daω arethe input, and αa is the output, and then compute a tentative point ωa by callingProcedure ωk = updateSolution(ωa, daω, α

a) where ωk, dω, and αa are the input andωa is the output.

Mehrotra suggested to determine the barrier parameters γa and ηa by using thepredicted reduction in the complementarity gap µ(ωa) along the affine direction. Theidea behind this heuristic is to reduce the centrality parameter γa more when the IPMcan make good progress towards optimality in the affine direction. In our implemen-tation we compute the barrier parameters γa and ηa as follows:

γa = min

0.99,max

(µ(ωa)µ(ωk)

)3

, 0.01

,

ηa = 1− γa.

The new complementarity gap is given by

(Xk +Dax)(sk + das) =Da

xdas ,

(Zk +Daz )(yk + day) =Da

zday,

(τk + daκ)(κk + daτ ) = daτdaκ,

where Dax := diag(dax) and Da

z := diag(daz). Since the new complementarity productideally should be equal to µ(ωa)e, the new search direction compensates for the errorand thus is computed with the following right-hand side:

rp :=−ηarvp, rxs := γaµ(ωa)e− Xksk −Daxdas ,

rd :=−ηarvd, rzy := γaµ(ωa)e− Zkyk −Dazday,

rg :=−ηarvg, rτκ := γaµ(ωa)− τkκk − daτdaκ.

The resulting direction here is called “Mehrotra direction” and is denoted by dmω .

4.5. Search guided by the potential function. The implementation logicin iOptimize that combines the predictor and corrector search directions at eachiteration is now given. At the core of our implementation is to ensure sufficientreduction in the potential function at each iteration, while using Mehrotra directionto further improve reduction in the potential function.

At each iteration k we first check if Φn+m+1(ωk) ≤ ι(n+m+1) log(n+m+1). Thissafeguard is used to make sure that our algorithm generates a maximal complementarysolution. If it is not satisfied, then we use the theoretical direction computation phase;otherwise we compute a Mehrotra direction dmω and use it to generate a tentative step


size ωm by calling Procedure αm =computeStepSize(ωk, dmω ), where ωk and dmω arethe input, and αm is the output. Then we compute a tentative point ωm by callingProcedure ωm = updateSolution(dmω , ω

k, αm), where ωk, dmω , and αm are the inputand ωm is the output. Though dmω generally enhances the convergence, it does notguarantee reduction of the potential function value, or maintain the overall decreasein the infeasibility to complementarity ratio (Assumption 3.1). Hence we determinethe quality of the Mehrotra direction by evaluating the potential function value at ωm

and check if ωm satisfies Assumption 3.1. We note that checking 3.1(ii) requires theknowledge of the scaled Lipschitz constant, which may not be available in practice;hence, we simply check if µ(ωm) < µ(ωk). If Φρ(ω

m) < Φρ(ωk) and Assumption 3.1

is satisfied, we set ωk+1 := ωm; otherwise we compute a theoretical direction dtω from(4.5) and (4.6), and iteratively employs a golden-section linear search method [13] tofind a controllable step size αt, and use it to compute a new tentative point ωt bycalling Procedure ωt =updateSolution(ωk, dtω, α), where ωk, dtω, and αt are the inputand ωt is the output. We terminate the line search if the potential function achievessufficient decrease.

Note that if the scaled Lipschitz constant is large, then many function evaluationsmay be needed in the golden section line search. To avoid this we simply take the stepafter 4 iterations of golden section method, and monitor the progress of the potentialfunction over multiple interior point iterates. We ensure that the potential functionis eventually reduced over these multiple iterates.

A flowchart of the implementation of search direction computations is given inFigure 4.1.

4.6. Starting point and termination criteria. Andersen and Ye [11] showthat the HMCP IPM achieves consistently good performance even for a trivial startingpoint

(x0; x0; y0; y0; τ0; z0; s0;κ0) = (e; 0; e; 0; 1; e; e; 1).

This trivial starting point is used in the results reported in this paper. However, itmight be possible to further improve the computational results by implementing a(better) non-trivial starting point.

Another important issue is the termination criteria. We choose the terminationcriteria used by Andersen and Ye [11] with slight modifications. Define the primaland dual objective values by

PriObj := c(x/τ), and DualObj := L(x/τ, y/τ)−∇xL(x/τ, y/τ)(x/τ).

Let rvkp , rvkd , and rvkg be the vectors of the residuals corresponding to the kth iterate.

We terminate the algorithm whenever one of the following criteria is satisfied:• For QP, an (approximate) optimal solution is obtained if

‖rvkp‖τ 1 + ‖b‖

< 10−8,

‖rvkd‖τ 1 + ‖c‖

< 10−8,

min

|DualObj− PriObj|

1 + |DualObj|,|rvkg |

1 + |rv0g |

< 10−8.


compute Mehrotra

direction dm

from (4.5) and

(4.6) with RHS (4.10)

call computeStepSize

αm

=(ωk, d

m)

call updateSolution

ωm

=(ωk, d

m, α

m)

compute centering

direction d t

from (4.6) and

(4.7) with RHS γ=η=0.5

Φρ(ωm

) decreases

and Assumption 3.1

satisfied?

Iteratively compute

step size αt

by golden-section line

search method

call updateSolution

ωt=(ω

k, d

t, α

t)

No

Set ωk+1

:=ωm

Input: ωk

theoretical direction computations

Potential

function value of

ωt decreases?

No

Set ωk+1

:=ωt

Output: ωk+1

golden-section line search method

Line search max

iterations reached?

No

Yes

YesYes

No

Yes

Φn+m+1(ωk) ≤

ι(n+m+1)log(n+m+1)

?

Fig. 4.1: Implementation of search direction computations

• For QCQP and CP, an (approximate) optimal solution is obtained if

‖rvkp‖max

1, ‖rv0

p‖ < 10−8,

‖rvkd‖max 1, ‖rv0

d‖< 10−8,

|rvkg |max

1, |rv0

g | < 10−6.

• For QP, QCQP, and CP, the problem is (near) infeasible if

µ(ωk) < µ(ω0)10−8 andτk

min 1, κk<τ0

κ010−12.

We note that Andersen [9] developed an infeasibility certificate for the homoge-neous monotone complementarity problems, that provide further information aboutwhether the primal or dual problem is infeasible. This feature, however, has not beenimplemented in the current version of iOptimize.

5. Computational results. iOptimize was compiled with the Visual Studio2010 compiler. All computations were performed on a 3.2 GHz Intel Dual-Core CPUmachine with 4GB RAM. The program is run on one CPU only. An AMPL [1] interface


Parameter Value Explanation of the parameter

θ n+ m+ 1 Parameter in the potential function (3.1)ρ 100 Parameter in the potential function (3.1)(θl, θu) (0.999, 1.0) Parameters for QP in Procedure computeStepFactor(θl, θu) (0.9, 1.0) Parameters for QCQP and CP in Procedure computeStepFactorι (n+ m+ 1)2 Parameter in (3.28)

(Θ(1),Ω(1)) (10−2, 102) Parameters in Assumption 3.1

Table 5.1: The default parameter settings in iOptimize

is implemented in iOptimize to read the AMPL nonlinear models and the correspondingfirst and second-derivatives. All problems are solved using the same default parametersettings, which are shown in Table 5.1.

To provide a clearer summary of the results, we use performance profiles [15].Consider a set A of na algorithms, a set P of np problems, and a performance measurema,p, e.g., node count or computation time. We compare the performance on problemp by algorithm a with the best performance by any algorithm on this problem usingthe following performance ratio

rp,a =mp,a

min mp,a | a ∈ A.

We therefore obtain an overall assessment of the performance of the algorithm bydefining the following value

ρa(τ) =1

npsize p ∈ P | rp,a ≤ τ .

This represents the probability for algorithm a that the performance ratio rp,a iswithin a factor τ of the best possible ratio. The function ρa(·) represents the distri-bution function for the performance ratio.

5.1. Computational results on feasible problems. We solve the convex QPproblems from Maros and Mezaros’s QP test set [24], the convex QCQP problems fromMittelmann’s QCQP test set [5] and Vanderbei’s AMPL nonlinear Cute and Non-Cute

set [2], the general convex CP problems from Vanderbei’s AMPL nonlinear test set [2],and from Leyffer’s mixed-integer nonlinear test set [6] while ignoring the integralityrequirements.

Tables A.1–A.4 give the computational results that are obtained with the defaultparameter settings. Each row in these tables contains the profiles of the problems be-fore and after presolving, the primal and the dual objectives, the number of iterations(Avg Iters), the computation times (Presolve Time and Solver Time), the propor-tion of the effort in evaluating the potential function (Prop Φ (%)), and the averagenumber of search directions used (Avg Dirs) (here two means iOptimize uses pureNewton direction and Mehrotra corrector, whereas one means iOptimize uses thetheoretical direction only). We note that the QCQP problems in Mittelmann’s testset are equivalent to those of Maros and Mezaros’s QP test set, as each QP problemis reformulated as a QCQP problem with an additional quadratic constraint and afree variable.

Tables A.5–A.8 provide a comparison between MOSEK homogeneous and self-dualinterior point optimizer and iOptimize under their default settings. We focus on thenumber of iterations needed to achieve the desired accuracy by both solvers.


Table A.5 gives the computational results for solving 127 QP problems. Theaverage number of iterations used by iOptimize (15.92 iters) and MOSEK (16.22 iters)are comparable. All problems can be solved within 50 iterations by both solvers. Notethat for three problems (liswet4, powell20, and qpilotno) MOSEK terminates withNEAR OPTIMAL status. This implies MOSEK cannot compute a solution that has theprescribed accuracy.

Table A.6 gives the computational results for solving Mittelmann’s QCQP test setproblems. Here two different average performance results are provided. “Avg1” pro-vides the average performance on problems for which MOSEK terminate with OPTIMAL,NEAR OPTIMAL or UNKNOWN status. For problems QQ-aug2dqp and QQ-powell20,MOSEK is not able to converge to the optimal solution within 400 iterations. We excludeboth problems in “Avg1”. For problems QQ-laser, QQ-qforplan, and QQ-qseba,MOSEK requires more than 50 interior point iterations to successfully solve. We recom-pute the average performance statistics again in “Avg2” by further excluding thesethree examples. Considering average performances in these experiments, iOptimize(Avg1=16.43 iters, Avg2=16.26 iters) is significantly better than MOSEK (Avg1=20.91iters, Avg2=18.34 iters). Note that for QQ-liswet1, QQ-liswet7–QQ-liswet12, MOSEKterminates with UNKNOWN status. This may happen when MOSEK converges slowly, andhence the primal, dual, or gap residuals may not attain the prescribed accuracy. Onthe other hand, iOptimize is able to terminate with OPTIMAL status for all the testproblems within 40 iterations.

Table A.7 contains the computational results for solving 11 QCQP problems inthe Vanderbei’s AMPL nonlinear test set. Comparing the size to those in Mittelmann’sQCQP test set, the problems here are relatively small, but may include a quadraticobjective term and more than one quadratic constraint. Considering the average iter-ations required for solving the problems, both solvers generate similar computationalresults (MOSEK Avg=11.36 iters versus iOptimize Avg=11.64 iters).

Table A.8 contains the computational results for solving 46 general CP prob-lems. Thirty five of the CP problems are selected from Vanderbei’s AMPL nonlin-ear Cute and Non-Cute set, and 11 of the CP problems are selected from Leyffer’smixed-integer nonlinear test set. Two different average performance statistics areprovided. “Avg1” provides the average performance on all the instances, exceptproblem dallass, for which MOSEK fails to converge to an optimal solution within400 iterations. For problems dallasm, dallasl, and elena-s383, MOSEK respec-tively requires 128, 74, and 85 iterations before terminating with OPTIMAL status.Thus we recompute the average statistics again in “Avg2” by additionally exclud-ing these three problems. Comparing “Avg1”, the average number of iterations usedby iOptimize (14.00 iters) is better than those used by MOSEK (18.46 iters). Byexcluding these problems for which MOSEK requires more than 50 iterations to solve(“Avg2”), the performance comparison between both solvers becomes comparable,though iOptimize (12.69 iters) still performs slightly better than MOSEK (13.07 iters).Note that for problems antenna-antenna2, firfilter-fir convex, and wbv-lowpass2,MOSEK terminates with NEAR OPTIMAL status.

The average number of directions used by iOptimize is ≈ 1.93, i.e., Mehrotradirection was not used in about 7% of the iterations. It implies that the averageproportion of the computations for the theoretical direction with a fixed step sizeis not significant. In these experiments we observed that Assumption 3.1 and therequirement Φn+m+1(ωk) ≤ ι(n+ m+ 1) log(n+ m+ 1) is always satisfied over allthe iterates, implying that iOptimize uses the theoretical direction only in the case


where Mehrotra direction fails to reduce the potential function value. We also notethat the average proportion of the efforts used for the potential function evaluations is3.1%, suggesting that the computational demands for evaluating the potential functionis also not significant, while a potential reduction algorithm is implemented thatguarantees global linear polynomial time convergence.

Figure 5.1 shows the performance profiles for feasible problems comparing theiteration performances of iOptimize and MOSEK.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Performance Profile − Iterations

τ

Pro

babi

lity

MosekiOptimize

(a) Maros and Mezaros’s QP feasible problems

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

0.4

0.5

0.6

0.7

0.8

0.9


τ

Pro

babi

lity

MosekiOptimize

(b) Mittelmann’s QCQP feasible problems

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95


τ

Pro

babi

lity

MosekiOptimize

(c) Vanderbei’s QCQP feasible problems

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30.4

0.5

0.6

0.7

0.8

0.9


τ

Pro

babi

lity

MosekiOptimize

(d) General CP feasible problems

Fig. 5.1: Performance profiles of iOptimize and MOSEK on feasible problems

5.2. Computational results on infeasible problems. In this section we testthe performance of iOptimize for solving the QCQP and CP infeasible problems.With the lack of the infeasible test set in the public domain, we create infeasible testproblems by adding invalid objective constraints to the feasible problems.

Let x∗ be an optimal solution of problem (4.1). We create a corresponding infea-sible problem that has the form:

min c(x)(5.1)

s.t. ai(x) ≥ 0, i = 1, . . . ,mn,

ai(x) = 0, i = m+ 1, . . . ,m,


c(x) ≤ c(x∗)− λ,x ≥ 0, and m is free,

where λ is a constant with positive value. We choose

λ :=

2|c(x∗)|, if |c(x∗)| ≥ 100,100, if |c(x∗)| < 100.

Hence, the problems are made significantly infeasible.Using this construction we create 127 infeasible QCQP problems from Maros

and Mezaros’s QP test set [24], 11 QCQP infeasible problems from Vanderbei’sAMPL nonlinear Cute and Non-Cute set [2], 35 CP infeasible problems from Van-derbei’s AMPL nonlinear Cute and Non-Cute set [2], and 11 CP infeasible problemsfrom Leyffer’s mixed-integer nonlinear test set [6] without integrality requirements.

Tables A.9–A.11 give the computational results that are obtained with the defaultparameter settings. The number of iterations (Avg Iters), the computation times (Pre-solve Time and Solver Time), the average number of search directions used (Avg Dirs),and the proportion of the effort in evaluating the potential function (Prop Φ (%)) aregiven. Tables A.12–A.14 provide a comparison between MOSEK and iOptimize undertheir default settings.

We focus on the number of iterations needed to detect infeasibility by bothsolvers. Table A.12 contains the computational results for solving 127 QCQP in-feasible problems created from Maros and Mezaros’s QP test set. For 38 problemsMOSEK fails to detect infeasibility within 400 iterations. For problems iQQ-qscagr25

and iQQ-qstandat, MOSEK terminates with UNKNOWN status. We compute the av-erage iterations in “Avg1” by excluding these 38 examples. Comparing “Avg1”,iOptimize (28.15 iters) requires approximately 30% fewer iterations than MOSEK (42.20iters). This is because MOSEK requires more than 60 iterations to detect the infeasibil-ity in 12 problems. We further exclude these 12 problems and recompute the averageiterations in “Avg2”. Comparing “Avg2”, the performance of iOptimize (28.15 iters)is comparable to that of MOSEK (27.39 iters). However, iOptimize is able to correctlydetect infeasibility for each the problem within 50 interior point iterations.

Table A.13 has the computational results for solving the 11 QCQP infeasibleproblems created from Vanderbei’s AMPL nonlinear test set. We provide the aver-age statistics by excluding problem ipolak4, which MOSEK fails to solve within 400iterations. Though the comparison shows that iOptimize (24.60 iters) requires ap-proximately 20% more iterations than MOSEK (20.30 iters), iOptimize correctly solvesall these infeasible problems.

Table A.14 has the computational results for solving 46 CP infeasible prob-lems created from Vanderbei’s AMPL nonlinear Cute and Non-Cute set, and Leyffer’smixed-integer nonlinear test set. For problems idallasl, idallasm, and idallass,MOSEK receives function evaluate error from AMPL. iOptimize receives the same er-ror while solving problem idallasl. In seven problems MOSEK fails to declare theinfeasible status within 400 iterations. We compute the average statistics “Avg2”by excluding these 10 problems. Comparing “Avg1”, MOSEK (66.33 iters) requiressignificantly more iterations than iOptimize (19.44 iters). This happens becauseMOSEK requires more than 60 iterations to detect infeasibility in 13 problems whereasiOptimize needs this in only one problem. We further exclude these 10 problems andrecompute the average statistics in “Avg2”. Comparing “Avg2”, iOptimize (16.04iters) requires 60% more iteration than MOSEK (11.74 iters). This is because MOSEK is


able to detect infeasibility during the preprocessing phase in 11 cases; on the otherhand, iOptimize preprocessor does not detect any infeasibility. This highlights theimportance of advanced preprocessing implementations, which is not the focus ofresearch described in the current paper.

Note that for problem antenna-iantenna2, iOptimize requires 186 iterationsto solve, which is significantly larger than the average iterations (28.07 iters). Witha fixed step factor parameter settings (θl, θu) = (0.9, 0.9), iOptimize can solve thisproblem in 46 iterations.

Observe that the average number of directions used for this problem is around1.39, which is significantly smaller than that used for solving the feasible problems.This suggests the importance of using the theoretical direction and potential functionto decide the usefulness of a direction.

Figure 5.2 shows the performance profiles for the infeasible problems compar-ing the iteration performances of iOptimize and MOSEK. Note that problems whichiOptimize or MOSEK fails to solve or solves them during the preprocessing phase arenot considered.

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95


τ

Pro

babi

lity

MosekiOptimize

(a) Mittelmann’s QCQP infeasible problems

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


τ

Pro

babi

lity

MosekiOptimize

(b) Vanderbei’s QCQP infeasible problems

1 2 3 4 5 6 7 8 9

0.4

0.5

0.6

0.7

0.8

0.9


τ

Pro

babi

lity

MosekiOptimize

(c) General CP infeasible problems

Fig. 5.2: Performance profiles of iOptimize and MOSEK on infeasible problems


5.3. Computational comparison with general nonlinear solvers. In thissection we compare iOptimize with general nonlinear solver Knitro and Ipopt onCP feasible and infeasible problems. The computational results on QP and QCQPproblems are not provided because these problems are stored in MOSEK QPS format,which Ipopt and Knitro do not support. Note that Knitro provides three differentalgorithms for solving the nonlinear problems. In our comparison, we choose the directprimal-dual IPM. For Ipopt, we use MA57 library for the sparse symmetric indefinitematrix factorization, which is the same in iOptimize. Default parameter settings areused for both solvers expect that maximum iterations allowed is set at 400.

We first focus on the computational results for solving the CP feasible problems.The last two columns of Table A.8 contain the number of iterations used by Ipopt andKnitro. Considering the average number of iterations used for solving these problems,iOptimize (12.77 iters) ranked first, Knitro (18.95 iters) ranked second, whereasIpopt (27.84 iters) ranked last. For problems dallasm and polak3, both Knitro andIpopt received a function evaluation error from AMPL. Ipopt failed to solve dallass

to optimality in 400 iterations. Observe that Ipopt requires more than 50 iterationsfor three problems and Knitro requires more than 50 iterations for five problems,whereas iOptimize solves all feasible CP problems within 50 iterations.

Now we focus on the CP infeasible problems. Computational results from Ipopt

and Knitro are reported in the last two columns of Table A.14. Knitro failed to solve13 problems within 400 iterations, whereas Ipopt fails in three problems. Knitro re-ceived an AMPL function evaluation error in three problems and Ipopt received thesame error in four problems. Note that both Knitro and Ipopt terminate with a nearoptimal status in one problem. In three problems Ipopt reported error in feasibilityrestoration phase. Overall, Ipopt successfully detects the infeasibility in 35 problemsand Knitro in 29 problems.

6. Concluding remarks. In this paper we have extended Mehrotra-Huang po-tential function [26] to the context of MCP, and shown that our potential reductioninterior point method generates a maximal complementary solution with desired ac-curacy if a scaled Lipschitz condition is satisfied. However, the main emphasis ofthe analysis is not to improve the best known complexity, but to make sure the inte-rior point method maintains the global linear polynomial time convergence propertieswhile achieving practical performance. We have implemented an interior point solverin a software package named iOptimize that does not ignore the theory of potentialfunction in the context of interior point methods, while adding only a small addi-tional computational burden. Computational comparisons on feasible and infeasibletest problems show that, in terms of iteration counts and the number of problemssolved, iOptimize appears to perform more efficiently and robustly than a maturecommercial solver MOSEK. We also find that iOptimize seems to detect infeasibilitymore reliably than general nonlinear solvers Ipopt and Knitro. It might be possibleto further improve the computational results by implementing a better preprocessor,a non-trivial starting point, or better step size computations. However, that was notthe focus of the research reported here.

At the core of the solver is the matrix factorization method, for which iOptimize

uses Bunch-Parlett factorization [14]. The method used in MOSEK is not known to us.However, we point out that numerical instability may result if quasi-definite systemapproach is used [11, 37]. Bunch-Parlett factorization in general is numerically stabler,but requires additional computational effort.

We also implemented a version of Gondzio’s higher-order correction strategy [18]


in iOptimize. Our implementation of this strategy is a direct extension of the ap-proach in Mehrotra and Huang [26]. Computational results show that iOptimize us-ing Mehrotra direction and up to two additional Gondzio’s correctors for the QPproblems saves only approximately 9% iterations. However, we did not see any im-provement on QCQP and CP problems. The practical value of using other correctionstrategies is a topic of future research.

We note that the computational performance for a problem can be improvedpossibly with different parameter settings. For example, with parameters (θl, θu) =(0.99, 1), the average iterations for solving the QCQP and CP feasible problems is re-duced further by about 6%. However, this number is increased by about 9% while solv-ing the QCQP and CP infeasible problems. In fact, for antenna-iantenna vareps,iOptimize under default setting ((θl, θu) = (0.9, 1)) needs only 47 iterations to cor-rectly detect infeasibility whereas with parameters (θl, θu) = (0.99, 1) it will require454 iterations. A very aggressive step to the boundary is not desirable for infeasibleand general convex programming problems.

Finally, we note that a small but difficult two-variable example can be foundat http://mosek.com/fileadmin/talks/func versus conic.pdf, (page 29). Withthe default setting iOptimize uses 24 iterations to solve this example to optimalitywhereas MOSEK with default setting uses 124 iterations.

REFERENCES

[1] AMPL: A modeling language for mathematical programming. www.ampl.com.[2] AMPL nonlinear test problems. http://www.orfe.princeton.edu/~rvdb/ampl/nlmodels/.[3] COIN-OR Ipopt interior point optimizer. http://www.coin-or.org/ipopt/.[4] Knitro: A aolver for nonlinear optimization. http://www.ziena.com/knitro.html.[5] Mittelmann’s convex QCQP test problems. http://plato.asu.edu/ftp/ampl files/qpdata ampl/.[6] Mixed integer nonlinear programming (minlp) test problems.

http://wiki.mcs.anl.gov/leyffer/index.php/MacMINLP.[7] MOSEK: A high performance solftware for large-scale optimization problems.

http://www.mosek.com/.[8] Netlib. http://www.netlib.org/lp/index.html.[9] E.D. Andersen, On primal and dual infeasibility certificates in a homogeneous model for

convex optimization, SIAM Journal On Optimization, 11 (2000), pp. 380–388.[10] E.D. Andersen, C. Roos, and T. Terlaky, On implementing a primal-dual interior-point

method for conic quadratic optimization, Mathematical Programming, 95 (2003), pp. 249–277.

[11] E.D. Andersen and Y. Ye, A computational study of the homogeneous algorithm for large-scale convex optimization, Computational Optimization and Applications, 10 (1998),pp. 243–269.

[12] , On a homogeneous algorithm for the monotone complementarity problem, MathematicalProgramming, 84 (1999), pp. 375–399.

[13] M.S. Bazarra, H.D. Sherali, and C.M. Shetty, Nonlinear Programming: Theory And Al-gorithms, John Wiley & Sons, New York, 2 ed., 1993.

[14] J.R. Bunch and B.N. Parlett, Direct methods for solving symmetric indefinite systems oflinear equations, SIAM Journal on Numerical Analysis, 8 (1971), pp. 639–655.

[15] E. Dolan and J. More, Benchmarking optimization software with performance profiles, Math-ematical Programming, 91 (2002), pp. 201–213.

[16] I.S. Duff, MA57–a code for the solution of sparse symmetric definite and indefinie systems,ACM Transactions on Mathematical Software, 30 (2004), pp. 118–144.

[17] D. Goldfarb and S. Mehrotra, A relaxed variant of Karmarkar’s algorithm, MathematicalProgramming, 40 (1988), pp. 285–315.

[18] J. Gondzio, Multiple centrality corrections in a primal-dual method for linear programming,Computational Optimization and Applications, 6 (1996), pp. 137–156.

[19] O. Guler, Existence of interior points and interior paths in nonlinear monotone complemen-tarity problems, Mathematics of Operations Research, 18 (1993), pp. 148–162.


[20] O. Guler and Y. Ye, Convergence behavior of interior-point algorithms, mathematical Pro-gramming, 60 (1993), pp. 215–228.

[21] N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, 4(1984), pp. 373–395.

[22] M. Kojima, S. Mizuno, and A. Yoshise, An O(√nL) iteration potential reduction algorithm

for linear complementary problems, Mathematical Programming, 50 (1991), pp. 331–342.[23] Z.-Q. Luo, J.F. Sturm, and S. Zhang, Conic convex programming and seld-dual embedding,

Optimization Method and Software, 14 (2000), pp. 169–218.[24] I. Maros and C. Mezaros, A repository of convex quadratic programming problems, Opti-

mization Methods and Software, 11&12 (1999), pp. 671–681.[25] S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM Journal

on Optimization, 2 (1992), pp. 575–601.[26] S. Mehrotra and K.-L. Huang, Computational experience with a modified potential reduction

algorithm for linear programming, Optimization Method and Software, 27 (2012), pp. 865–891.

[27] M.D.C. Monteiro and I. Adler, An extension of Karmarkar type algorithm to a class ofconvex separable programming problems with global rate of convergence, Mathematical ofOperations Research, 15 (1990), pp. 408–422.

[28] Y.E. Nesterov and M.J. Todd, Self-scaled barriers and interior-point methods for convexprogramming, Mathematics of Operations Research, 22 (1997), pp. 1–42.

[29] F. Porta and Y. Ye, Interior-point methods for nonlinear complementarity problem, Journalof Optimization Theory and Applications, 68 (1996), pp. 617–642.

[30] F.A. Potra and Y. Ye, A quadratically convergent polynomial algorithm for solving entropyoptimization problems, SIAM Jorunal on Optimization, 3 (1993), pp. 843–860.

[31] , Interior point methods for nonlinear complementarity problems, Journal of Optimiza-tion Theory and Application, 88 (1996), pp. 617–647.

[32] R.T. Rockafellar, Convex Analysis, Princeton University Press, 1996.[33] A. Skajaa, J.B. Jørgensen, and P.C. Hansen, On implementing a ho-

mogeneous interior-point algorithm for nonsymmetric conic optimization,http://www.imm.dtu.dk/English/Service/

Phonebook.aspx?lg=showcommon&id=f30a529b-4c6b-4642-a67b-cca4f74d0a3a, (2011).[34] J.F. Sturm, Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,

Optimization Method and Software, 11–12 (1999), pp. 625–653.[35] K. Tanabe, Centered newton method for mathematical programming, System Modelling and

Optimization, 113 (1988), pp. 197–206.[36] M.J. Todd and Y. Ye, A centered projective algorithm for linear programming, Mathematics

of Operations Research, 15 (1990), pp. 508–529.[37] R.J. Vanderbei, Symmetric quasi-definite matrices, SIAM Journal on Optimization, 5 (1995),

pp. 100–113.[38] X. Xu, P. Hung, and Y. Ye, A simplified homogeneous and self-dual linear programming

algorithm and its implementation, Annals of Operations Research, 62 (1996), pp. 151–171.[39] Y. Ye, M.J. Todd, and S. Mizuno, An O(

√nL) -iteration homogeneous and self-dual linear

programming algorithm, Mathematics of Operations Research, 19 (1994), pp. 53–67.[40] S. Zhang, A new self-dual embedding method for convex programming, Journal of Global

Optimization, 29 (2004), pp. 479–496.[41] J. Zhu, A path following alrogirhm for a class of convex programming problems, Method and

Models of Operations Research, 36 (1992), pp. 359–377.

Appendix. A.


Befo

reP

reso

lvin

gA

fter

Pre

solv

ing

Prim

al

Dual

Pre

solv

eSolv

er

Avg

Avg

Pro

pΦ

Pro

ble

mR

ow

sC

ols

Row

sC

ols

Ob

jectiv

eO

bje

ctiv

eT

ime

Tim

eIte

rsD

irs(%

)

aug2d

10000

20200

9604

19800

1.6

8741175E

+06

1.6

8741175E

+06

0.0

33.9

12

1.3

34.3

9aug2dc

10000

20200

10000

20200

1.8

1836807E

+06

1.8

1836807E

+06

0.0

24.0

712

1.3

34.4

8aug2dcqp

10000

20200

10000

20200

6.4

9813474E

+06

6.4

9813473E

+06

0.0

22.4

13

26.8

1aug2dqp

10000

20200

10000

20200

6.2

3701204E

+06

6.2

3701200E

+06

0.0

12.6

414

27.5

aug3d

1000

3873

512

2673

5.5

4067726E

+02

5.5

4067726E

+02

0.0

10.2

78

1.5

3.8

9aug3dc

1000

3873

1000

3873

7.7

1262439E

+02

7.7

1262439E

+02

0.0

10.3

88

1.5

4.3

8aug3dcqp

1000

3873

1000

3873

9.9

3362148E

+02

9.9

3362145E

+02

0.0

10.3

611

24.8

4aug3dqp

1000

3873

1000

3873

6.7

5237672E

+02

6.7

5237670E

+02

0.0

10.3

913

25.5

9cont-0

50

2401

2597

2401

2597

-4.5

6385090E

+00

-4.5

6385090E

+00

0.0

10.6

511

23.4

9cont-1

00

9801

10197

9801

10197

-4.6

4439787E

+00

-4.6

4439787E

+00

0.0

14.5

511

22.1

1cont-1

01

10098

10197

10098

10197

1.9

5527325E

-01

1.9

5527325E

-01

0.0

14.0

69

21.9

5cont-2

00

39601

40397

39601

40397

-4.6

8487592E

+00

-4.6

8487592E

+00

0.0

443.3

411

20.9

9cont-2

01

40198

40397

40198

40397

1.9

2483373E

-01

1.9

2483373E

-01

0.0

442.0

910

1.9

1cont-3

00

90298

90597

90298

90597

1.9

1512286E

-01

1.9

1512286E

-01

0.0

8247.5

913

1.9

20.4

7cvxqp1

m500

1000

500

1000

1.0

8751157E

+06

1.0

8751157E

+06

00.4

110

22.2

5cvxqp1

s50

100

50

100

1.1

5907181E

+04

1.1

5907181E

+04

00.0

88

21.3

2cvxqp2

m250

1000

250

1000

8.2

0155431E

+05

8.2

0155431E

+05

00.2

310

22.7

6cvxqp2

s25

100

25

100

8.1

2094048E

+03

8.1

2094048E

+03

00.1

10

1.9

1.0

8cvxqp3

m750

1000

750

1000

1.3

6282874E

+06

1.3

6282874E

+06

00.6

512

21.4

1cvxqp3

s75

100

75

100

1.1

9434322E

+04

1.1

9434322E

+04

00.0

910

20

dpklo

177

133

77

133

3.7

0096217E

-01

3.7

0096217E

-01

00.0

43

20

dto

c3

9998

14999

9997

14996

2.3

5262481E

+02

2.3

5262481E

+02

0.0

20.9

34

28.7

5dual1

185

185

3.5

0129664E

-02

3.5

0129651E

-02

00.1

12

22.0

6dual2

196

196

3.3

7336761E

-02

3.3

7336761E

-02

00.0

89

20

dual3

1111

1111

1.3

5755838E

-01

1.3

5755833E

-01

00.0

99

21.2

5dual4

175

175

7.4

6090842E

-01

7.4

6090840E

-01

00.0

810

21.3

2dualc

1215

223

13

21

6.1

5525083E

+03

6.1

5525083E

+03

00.2

625

1.4

80

dualc

2229

235

915

3.5

5130769E

+03

3.5

5130769E

+03

00.2

120

1.5

50

dualc

5278

285

18

4.2

7232327E

+02

4.2

7232327E

+02

0.0

10.1

213

1.6

20

dualc

8503

510

15

22

1.8

3093588E

+04

1.8

3093588E

+04

0.0

10.2

19

1.6

30.5

genhs2

88

10

810

9.2

7173692E

-01

9.2

7173694E

-01

0.0

10.0

33

20

gould

qp2

349

699

349

699

1.8

4275300E

-04

1.8

4273273E

-04

00.1

613

23.2

5gould

qp3

349

699

349

699

2.0

6278397E

+00

2.0

6278397E

+00

0.0

10.1

411

24.5

8hs1

18

17

32

17

32

6.6

4820450E

+02

6.6

4820450E

+02

0.0

10.0

910

20

hs2

11

31

3-9

.99599996E

+01

-9.9

9600001E

+01

0.0

10.0

89

20

hs2

68

510

510

2.5

8580312E

-05

-5.0

6859105E

-05

00.1

12

20

hs3

51

41

41.1

1111111E

-01

1.1

1111111E

-01

00.0

55

20

hs3

5m

od

14

13

2.5

0000007E

-01

2.4

9999989E

-01

00.0

911

20

hs5

13

53

50.0

0000000E

+00

9.9

7371075E

-11

0.0

10.0

33

20

hs5

23

53

55.3

2664756E

+00

5.3

2664757E

+00

00.0

43

20

hs5

33

53

54.0

9302326E

+00

4.0

9302302E

+00

0.0

10.0

44

20

hs7

63

73

7-4

.68181818E

+00

-4.6

8181818E

+00

00.0

55

20

ksip

1001

1021

1001

1021

5.7

5797941E

-01

5.7

5797941E

-01

0.0

10.3

816

23.4

5

SOLVING CONVEX PROGRAMMING USING INTERIOR POINT METHOD 35B

efo

reP

reso

lvin

gA

fter

Pre

solv

ing

Pri

mal

Dual

Pre

solv

eSolv

er

Avg

Avg

Pro

pΦ

Pro

ble

mR

ow

sC

ols

Row

sC

ols

Ob

jecti

ve

Ob

jecti

ve

Tim

eT

ime

Iters

Dir

s(%

)

lase

r1000

2002

1000

2002

2.4

0960135E

+06

2.4

0960134E

+06

0.0

10.2

110

24.0

4lisw

et1

10000

20002

10000

20002

3.6

1224021E

+01

3.6

1224021E

+01

0.0

33.2

119

1.8

97.5

3lisw

et2

10000

20002

10000

20002

2.4

9980788E

+01

2.4

9980759E

+01

0.0

22.6

817

27.3

2lisw

et3

10000

20002

10000

20002

2.5

0012273E

+01

2.5

0012181E

+01

0.0

22.5

516

27.3

9lisw

et4

10000

20002

10000

20002

2.5

0001164E

+01

2.5

0001107E

+01

0.0

22.7

618

28.0

6lisw

et5

10000

20002

10000

20002

2.5

0343102E

+01

2.5

0341783E

+01

0.0

22.5

416

27.9

1lisw

et6

10000

20002

10000

20002

2.4

9957529E

+01

2.4

9957445E

+01

0.0

23

18

27.8

2lisw

et7

10000

20002

10000

20002

4.9

8840891E

+02

4.9

8840891E

+02

0.0

23.7

119

1.8

97.3

1lisw

et8

10000

20002

10000

20002

7.1

4470067E

+02

7.1

4470059E

+02

0.0

25.4

737

1.7

87.7

6lisw

et9

10000

20002

10000

20002

1.9

6325126E

+03

1.9

6325126E

+03

0.0

25.1

231

1.6

57.1

8lisw

et1

010000

20002

10000

20002

4.9

4858035E

+01

4.9

4857840E

+01

0.0

23.1

821

1.9

57.6

lisw

et1

110000

20002

10000

20002

4.9

5239659E

+01

4.9

5239569E

+01

0.0

23.2

322

1.9

57.8

2lisw

et1

210000

20002

10000

20002

1.7

3692743E

+03

1.7

3692743E

+03

0.0

24.8

932

1.7

57.6

lots

chd

712

712

2.3

9841590E

+03

2.3

9841589E

+03

00.0

59

20

mosa

rqp1

700

3200

700

3200

-9.5

2875441E

+02

-9.5

2875445E

+02

00.2

19

25.5

mosa

rqp2

600

1500

600

1500

-1.5

9748211E

+03

-1.5

9748212E

+03

00.1

49

27.3

5p

ow

ell20

10000

20000

10000

20000

5.2

0895828E

+10

5.2

0895828E

+10

0.0

21.4

68

1.8

87.3

9pri

mal1

85

410

85

410

-3.5

0129646E

-02

-3.5

0129670E

-02

00.1

19

22.9

7pri

mal2

96

745

96

745

-3.3

7336714E

-02

-3.3

7336763E

-02

00.1

72

1.0

4pri

mal3

111

856

111

856

-1.3

5755836E

-01

-1.3

5755837E

-01

00.2

19

24.7

1pri

mal4

75

1564

75

1564

-7.4

6090837E

-01

-7.4

6090843E

-01

00.2

92

5.0

8pri

malc

19

239

9239

-6.1

5525083E

+03

-6.1

5525083E

+03

00.1

418

1.8

31.4

9pri

malc

27

238

7237

-3.5

5130769E

+03

-3.5

5130769E

+03

00.2

25

1.6

83.9

6pri

malc

58

295

8206

-4.2

7232327E

+02

-4.2

7232327E

+02

00.0

710

1.9

1.4

3pri

malc

88

528

8528

-1.8

3094298E

+04

-1.8

3094298E

+04

00.1

512

1.8

33.3

6q25fv

47

820

1876

790

1846

1.3

7444479E

+07

1.3

7444478E

+07

0.0

21.5

527

24.4

8qadlitt

le56

138

55

137

4.8

0318859E

+05

4.8

0318859E

+05

00.0

711

20

qafiro

27

51

27

51

-1.5

9078179E

+00

-1.5

9078179E

+00

00.0

814

1.8

60

qbandm

305

472

252

419

1.6

3523420E

+04

1.6

3523420E

+04

00.1

720

20.6

qb

eaconf

173

295

107

219

1.6

4712060E

+05

1.6

4712060E

+05

00.1

14

21.9

6qb

ore

3d

233

334

123

225

3.1

0020080E

+03

3.1

0020080E

+03

00.1

719

1.7

92.4

5qbra

ndy

220

303

136

246

2.8

3751149E

+04

2.8

3751149E

+04

00.1

317

20

qcapri

271

482

243

438

6.6

7932932E

+07

6.6

7932932E

+07

00.3

736

1.7

22.7

2qe226

223

472

214

463

2.1

2653433E

+02

2.1

2653433E

+02

00.1

716

22.4

8qeta

macr

400

816

334

669

8.6

7603696E

+04

8.6

7603693E

+04

00.4

932

22.2

9qff

fff8

0524

1028

322

826

8.7

3147464E

+05

8.7

3147456E

+05

00.3

925

1.9

23.3

7qfo

rpla

n161

492

122

450

7.4

5663147E

+09

7.4

5663144E

+09

00.3

931

1.7

12.8

6qgfr

dxpn

616

1160

590

1134

1.0

0790585E

+11

1.0

0790585E

+11

00.3

930

1.9

3.3

9qgro

w15

300

645

300

645

-1.0

1693640E

+08

-1.0

1693640E

+08

00.2

822

22.2

1qgro

w22

440

946

440

946

-1.4

9628953E

+08

-1.4

9628953E

+08

00.4

326

23.7

7qgro

w7

140

301

140

301

-4.2

7987139E

+07

-4.2

7987139E

+07

00.1

821

23.9

1qis

rael

174

316

174

316

2.5

3478378E

+07

2.5

3478376E

+07

00.2

828

1.7

51.8

3qp

cble

nd

74

114

74

114

-7.8

4254164E

-03

-7.8

4254440E

-03

00.1

117

21.9

4


Befo

reP

reso

lvin

gA

fter

Pre

solv

ing

Prim

al

Dual

Pre

solv

eSolv

er

Avg

Avg

Pro

pΦ

Pro

ble

mR

ow

sC

ols

Row

sC

ols

Ob

jectiv

eO

bje

ctiv

eT

ime

Tim

eIte

rsD

irs(%

)

qp

cb

oei1

351

726

343

718

1.1

5039140E

+07

1.1

5039140E

+07

00.2

722

22.2

6qp

cb

oei2

166

305

126

265

8.1

7196226E

+06

8.1

7196221E

+06

00.2

226

1.8

81.8

3qp

csta

ir356

614

356

532

6.2

0438749E

+06

6.2

0438744E

+06

00.3

225

1.7

61.9

2qpilo

tno

975

2446

866

2151

4.7

2858690E

+06

4.7

2858690E

+06

0.0

11.8

442

1.8

62.8

qre

cip

e91

204

76

141

-2.6

6615999E

+02

-2.6

6616001E

+02

00.1

219

20.8

8qpte

st2

42

44.3

7187500E

+00

4.3

7187500E

+00

00.0

36

20

qsc

205

205

317

203

315

-5.8

1395349E

-03

-5.8

1395349E

-03

00.1

217

21.6

9qsc

agr2

5471

671

469

669

2.0

1737938E

+08

2.0

1737938E

+08

00.2

222

1.9

54.1

3qsc

agr7

129

185

127

183

2.6

8659486E

+07

2.6

8659485E

+07

00.1

422

1.8

22.1

3qsc

fxm

1330

600

311

581

1.6

8826916E

+07

1.6

8826916E

+07

00.2

925

22.7

9qsc

fxm

2660

1200

622

1162

2.7

7761616E

+07

2.7

7761615E

+07

00.4

933

23.5

6qsc

fxm

3990

1800

933

1743

3.0

8163545E

+07

3.0

8163545E

+07

00.7

335

1.9

43.3

7qsc

orp

io388

466

355

436

1.8

8050955E

+03

1.8

8050955E

+03

00.1

514

1.9

32.8

qsc

rs8490

1275

445

1230

9.0

4560015E

+02

9.0

4560013E

+02

0.0

10.4

124

1.9

62.7

qsc

sd1

77

760

77

760

8.6

6666669E

+00

8.6

6666667E

+00

00.1

59

22.0

5qsc

sd6

147

1350

147

1350

5.0

8082139E

+01

5.0

8082139E

+01

00.2

914

22.2

qsc

sd8

397

2750

397

2750

9.4

0763578E

+02

9.4

0763570E

+02

0.0

10.3

811

23.7

6qsc

tap1

300

660

284

644

1.4

1586111E

+03

1.4

1586111E

+03

0.0

10.3

317

21.8

8qsc

tap2

1090

2500

1033

2443

1.7

3502651E

+03

1.7

3502649E

+03

0.0

10.5

113

22.4

5qsc

tap3

1480

3340

1408

3268

1.4

3875468E

+03

1.4

3875468E

+03

0.0

10.6

415

23.9

5qse

ba

515

1036

514

1033

8.1

4818004E

+07

8.1

4818004E

+07

0.0

10.6

426

1.9

21.7

5qsh

are

1b

117

253

112

248

7.2

0078318E

+05

7.2

0078318E

+05

00.4

729

1.7

61.9

6qsh

are

2b

96

162

96

162

1.1

7036917E

+04

1.1

7036917E

+04

00.2

517

20.4

1qsh

ell

536

1777

487

1475

1.5

7263685E

+12

1.5

7263684E

+12

0.0

10.9

838

1.7

91.7

6qsh

ip04l

402

2166

323

2104

2.4

2001553E

+06

2.4

2001553E

+06

0.0

10.3

814

22.9

3qsh

ip04s

402

1506

235

1356

2.4

2499367E

+06

2.4

2499367E

+06

0.0

10.3

114

24.2

6qsh

ip08l

778

4363

638

4247

2.3

7604062E

+06

2.3

7604062E

+06

0.0

11.1

214

24.7

4qsh

ip08s

778

2467

366

2079

2.3

8572885E

+06

2.3

8572885E

+06

0.0

10.4

114

1.9

33.3

1qsh

ip12l

1151

5533

784

5226

3.0

1887658E

+06

3.0

1887658E

+06

0.0

22.0

616

24.0

4qsh

ip12s

1151

2869

412

2190

3.0

5696225E

+06

3.0

5696225E

+06

0.0

10.7

317

23.1

3qsie

rra1227

2735

1212

2705

2.3

7504582E

+07

2.3

7504582E

+07

00.5

420

25.5

3qsta

ir356

614

356

532

7.9

8545276E

+06

7.9

8545275E

+06

00.3

623

1.8

72.2

9qsta

ndat

359

1274

346

855

6.4

1183839E

+03

6.4

1183839E

+03

00.2

113

21.9

6s2

68

510

510

2.5

8580312E

-05

-5.0

6859105E

-05

00.0

612

20

stadat1

3999

6000

3999

6000

-2.8

5268639E

+07

-2.8

5268640E

+07

0.0

11.0

312

24.5

stadat2

3999

6000

3999

6000

-3.2

6266649E

+01

-3.2

6266649E

+01

0.0

11.3

218

25.6

1sta

dat3

7999

12000

7999

12000

-3.5

7794529E

+01

-3.5

7794530E

+01

0.0

12.5

19

27.0

1ta

me

12

12

0.0

0000000E

+00

-4.1

9289224E

-10

00.0

43

20

ubh1

12000

18009

12000

17997

1.1

1600082E

+00

1.1

1600082E

+00

0.0

31.1

93

210.7

4yao

2000

4002

2000

4000

1.9

7704256E

+02

1.9

7704256E

+02

0.0

11.1

123

1.5

74.5

2zecevic

22

42

4-4

.12499999E

+00

-4.1

2500002E

+00

00.0

45

20

Avg

0.0

13.4

415.9

21.9

23.0

3

Tab

leA

.1:

iOptimize

perform

an

ceon

Maro

san

dM

eszaro

s’sQ

Ptest

prob

lems

SOLVING CONVEX PROGRAMMING USING INTERIOR POINT METHOD 37B

efo

reP

reso

lvin

gA

fter

Pre

solv

ing

Pri

mal

Dual

Pre

solv

eSolv

er

Avg

Avg

Pro

pΦ

Pro

ble

mR

ow

sC

ols

Row

sC

ols

Ob

jecti

ve

Ob

jecti

ve

Tim

eT

ime

Iters

Dir

s(%

)

QQ

-aug2d

10001

20201

10001

20201

1.6

8741182E

+06

1.6

8741175E

+06

0.0

13.7

58

1.7

54.1

4Q

Q-a

ug2dc

10001

20201

10001

20201

1.8

1836815E

+06

1.8

1836807E

+06

0.0

23.4

78

1.7

54.7

QQ

-aug2dcqp

10001

20201

10001

20201

6.4

9813474E

+06

6.4

9813474E

+06

0.7

56.7

919

1.6

85.1

6Q

Q-a

ug2dqp

10001

20201

10001

20201

6.2

3701203E

+06

6.2

3701202E

+06

0.7

36.7

21

1.7

15.3

8Q

Q-a

ug3d

1001

3874

1001

3874

5.5

4067753E

+02

5.5

4067816E

+02

0.0

10.5

65

22.0

3Q

Q-a

ug3dc

1001

3874

1001

3874

7.7

1262488E

+02

7.7

1262604E

+02

0.0

10.2

15

26.3

8Q

Q-a

ug3dcqp

1001

3874

1001

3874

9.9

3362146E

+02

9.9

3362144E

+02

0.0

40.4

411

25.7

4Q

Q-a

ug3dqp

1001

3874

1001

3874

6.7

5237669E

+02

6.7

5237664E

+02

0.0

30.4

312

26.3

1Q

Q-c

ont-

050

2402

2598

2402

2598

-4.5

6385143E

+00

-4.5

6385137E

+00

0.0

10.7

511

23.5

1Q

Q-c

ont-

100

9802

10198

9802

10198

-4.6

4439991E

+00

-4.6

4439983E

+00

0.0

15.5

912

22.2

QQ

-cont-

101

10099

10198

10099

10198

1.9

5527204E

-01

1.9

5527225E

-01

0.0

15.2

111

21.9

7Q

Q-c

ont-

200

39602

40398

39602

40398

-4.6

8487611E

+00

-4.6

8487609E

+00

0.0

549.4

313

21.3

2Q

Q-c

ont-

201

40199

40398

40199

40398

1.9

2483103E

-01

1.9

2483155E

-01

0.0

457.0

312

20.8

4Q

Q-c

ont-

300

90299

90598

90299

90598

1.9

1512063E

-01

1.9

1512092E

-01

0.1

261.3

12

20.4

3Q

Q-c

vxqp1

m501

1001

501

1001

1.0

8751109E

+06

1.0

8751098E

+06

0.0

10.3

810

22.1

6Q

Q-c

vxqp1

s51

101

51

101

1.1

5907177E

+04

1.1

5907177E

+04

00.0

58

24.1

7Q

Q-c

vxqp2

m251

1001

251

1001

8.2

0155390E

+05

8.2

0155375E

+05

00.2

210

22.8

6Q

Q-c

vxqp2

s26

101

26

101

8.1

2093991E

+03

8.1

2093970E

+03

00.0

59

20

QQ

-cvxqp3

m751

1001

751

1001

1.3

6282857E

+06

1.3

6282854E

+06

0.0

10.7

813

21.5

7Q

Q-c

vxqp3

s76

101

76

101

1.1

9434322E

+04

1.1

9434317E

+04

00.0

69

20

QQ

-dpklo

178

134

78

134

3.7

0096112E

-01

3.7

0096109E

-01

00.0

45

20

QQ

-dto

c3

9999

15000

9999

14998

2.3

5262505E

+02

2.3

5262481E

+02

0.3

12.2

27

1.7

15.9

7Q

Q-d

ual1

286

286

3.5

0125784E

-02

3.5

0123964E

-02

00.0

713

21.4

7Q

Q-d

ual2

297

297

3.3

7335240E

-02

3.3

7334725E

-02

00.0

712

21.4

5Q

Q-d

ual3

2112

2112

1.3

5755782E

-01

1.3

5755743E

-01

00.0

812

21.3

5Q

Q-d

ual4

276

276

7.4

6090320E

-01

7.4

6089988E

-01

00.0

610

21.9

2Q

Q-d

ualc

1216

224

216

224

6.1

5524425E

+03

6.1

5524190E

+03

00.1

421

23.6

2Q

Q-d

ualc

2230

236

230

236

3.5

5130765E

+03

3.5

5130761E

+03

00.1

217

22.6

8Q

Q-d

ualc

5279

286

279

286

4.2

7232103E

+02

4.2

7231914E

+02

00.0

79

24.7

6Q

Q-d

ualc

8504

511

504

511

1.8

3093532E

+04

1.8

3093473E

+04

00.1

313

26.5

6Q

Q-g

enhs2

89

11

911

9.2

7173702E

-01

9.2

7173694E

-01

00.0

35

20

QQ

-gould

qp2

350

700

350

700

1.8

4274493E

-04

1.8

4273733E

-04

00.1

515

24.7

6Q

Q-g

ould

qp3

350

700

350

700

2.0

6278287E

+00

2.0

6278325E

+00

00.1

210

24.2

4Q

Q-h

s118

18

33

18

33

6.6

4820445E

+02

6.6

4820439E

+02

00.0

511

20

QQ

-hs2

12

42

4-9

.99600032E

+01

-9.9

9600019E

+01

00.0

510

20

QQ

-hs2

68

611

611

-2.1

2146135E

-07

-7.6

9541657E

-07

00.0

715

20

QQ

-hs3

52

52

51.1

1111110E

-01

1.1

1111107E

-01

00.0

36

20

QQ

-hs3

5m

od

25

25

2.4

9999954E

-01

2.4

9999859E

-01

00.0

612

20

QQ

-hs5

14

64

64.2

4138339E

-08

3.6

0237236E

-08

00.0

25

24.7

6Q

Q-h

s52

46

46

5.3

2664757E

+00

5.3

2664757E

+00

00.0

25

20

QQ

-hs5

34

64

64.0

9302193E

+00

4.0

9302286E

+00

00.0

36

20

QQ

-hs7

64

84

8-4

.68181818E

+00

-4.6

8181818E

+00

00.0

36

20

QQ

-ksi

p1002

1022

1002

1022

5.7

5797941E

-01

5.7

5797940E

-01

0.0

10.2

913

23.4

2


Befo

reP

reso

lvin

gA

fter

Pre

solv

ing

Prim

al

Dual

Pre

solv

eSolv

er

Avg

Avg

Pro

pΦ

Pro

ble

mR

ow

sC

ols

Row

sC

ols

Ob

jectiv

eO

bje

ctiv

eT

ime

Tim

eIte

rsD

irs(%

)

QQ

-lase

r1001

2003

1001

2003

2.4

0960146E

+06

2.4

0960145E

+06

00.6

19

1.5

34.7

9Q

Q-lisw

et1

10001

20003

10001

20003

3.6

1224031E

+01

3.6

1224021E

+01

0.0

24.8

721

1.9

55.6

3Q

Q-lisw

et2

10001

20003

10001

20003

2.4

9980889E

+01

2.4

9980663E

+01

0.0

24.2

317

25.2

9Q

Q-lisw

et3

10001

20003

10001

20003

2.5

0012248E

+01

2.5

0012132E

+01

0.0

23.9

417

26.1

3Q

Q-lisw

et4

10001

20003

10001

20003

2.5

0001177E

+01

2.5

0001054E

+01

0.0

24.2

618

25.6

7Q

Q-lisw

et5

10001

20003

10001

20003

2.5

0342522E

+01

2.5

0342269E

+01

0.0

24.4

420

25.9

5Q

Q-lisw

et6

10001

20003

10001

20003

2.4

9957486E

+01

2.4

9957422E

+01

0.0

24.2

219

26.1

6Q

Q-lisw

et7

10001

20003

10001

20003

4.9

8841004E

+02

4.9

8840891E

+02

0.0

25.8

522

1.7

35.4

QQ

-liswet8

10001

20003

10001

20003

7.1

4470246E

+02

7.1

4470012E

+02

0.0

27.6

830

1.7

5.4

8Q

Q-lisw

et9

10001

20003

10001

20003

1.9

6325241E

+03

1.9

6325122E

+03

0.0

210.4

336

1.6

75

QQ

-liswet1

010001

20003

10001

20003

4.9

4858015E

+01

4.9

4857829E

+01

0.0

26.3

129

25.8

4Q

Q-lisw

et1

110001

20003

10001

20003

4.9

5239698E

+01

4.9

5239560E

+01

0.0

25.3

23

1.9

66.6

3Q

Q-lisw

et1

210001

20003

10001

20003

1.7

3692791E

+03

1.7

3692742E

+03

0.0

29.6

238

1.6

85.4

3Q

Q-lo

tschd

813

813

2.3

9841591E

+03

2.3

9841578E

+03

00.0

39

20

QQ

-mosa

rqp1

701

3201

701

3201

-9.5

2875445E

+02

-9.5

2875470E

+02

00.2

710

26.3

QQ

-mosa

rqp2

601

1501

601

1501

-1.5

9748212E

+03

-1.5

9748217E

+03

00.1

59

25.5

6Q

Q-p

ow

ell2

010001

20001

10001

20001

5.2

0895810E

+10

5.2

0895828E

+10

0.0

27.8

19

1.3

73.7

2Q

Q-p

rimal1

86

411

86

411

-3.5

0129655E

-02

-3.5

0129677E

-02

00.1

10

21.0

4Q

Q-p

rimal2

97

746

97

746

-3.3

7336759E

-02

-3.3

7336761E

-02

00.1

39

23.4

2Q

Q-p

rimal3

112

857

112

857

-1.3

5755836E

-01

-1.3

5755837E

-01

00.2

611

24.9

6Q

Q-p

rimal4

76

1565

76

1565

-7.4

6090838E

-01

-7.4

6090843E

-01

00.2

29

24.4

6Q

Q-p

rimalc

110

240

10

240

-6.1

5525075E

+03

-6.1

5525083E

+03

00.1

119

20.9

7Q

Q-p

rimalc

28

239

8239

-3.5

5130754E

+03

-3.5

5130761E

+03

00.0

917

20

QQ

-prim

alc

59

296

9296

-4.2

7232136E

+02

-4.2

7232268E

+02

00.0

710

23.1

3Q

Q-p

rimalc

89

529

9529

-1.8

3094193E

+04

-1.8

3094258E

+04

00.1

112

21.8

3Q

Q-q

25fv

47

821

1877

800

1856

1.3

7444358E

+07

1.3

7444267E

+07

0.0

21.7

428

1.9

33.1

6Q

Q-q

adlittle

57

139

56

138

4.8

0318851E

+05

4.8

0318850E

+05

00.0

512

20

QQ

-qafiro

28

52

28

52

-1.5

9078178E

+00

-1.5

9078188E

+00

00.0

414

24.6

5Q

Q-q

bandm

306

473

278

445

1.6

3523409E

+04

1.6

3523397E

+04

00.1

821

22.2

7Q

Q-q

beaconf

174

296

141

257

1.6

4711933E

+05

1.6

4711773E

+05

00.0

913

23.5

3Q

Q-q

bore

3d

234

335

200

303

3.1

0020162E

+03

3.1

0020194E

+03

00.1

819

23.3

9Q

Q-q

bra

ndy

221

304

141

251

2.8

3751074E

+04

2.8

3751029E

+04

00.1

617

22.6

1Q

Q-q

capri

272

483

262

457

6.6

7932279E

+07

6.6

7932452E

+07

0.0

10.3

32

1.6

93.3

6Q

Q-q

e226

224

473

223

472

2.1

2653431E

+02

2.1

2653417E

+02

00.1

517

21.3

9Q

Q-q

eta

macr

401

817

401

814

8.6

7601160E

+04

8.6

7601476E

+04

00.5

933

23.9

7Q

Q-q

ffff

f80

525

1029

491

995

8.7

3147438E

+05

8.7

3146951E

+05

00.4

926

24.0

2Q

Q-q

forp

lan

162

493

136

467

7.4

5661267E

+09

7.4

5660721E

+09

0.0

10.2

723

23.0

4Q

Q-q

gfrd

xpn

617

1161

591

1135

1.0

0790508E

+11

1.0

0790507E

+11

0.0

10.3

24

24.3

6Q

Q-q

gro

w15

301

646

301

646

-1.0

1693609E

+08

-1.0

1693652E

+08

00.2

621

22.7

6Q

Q-q

gro

w22

441

947

441

947

-1.4

9628886E

+08

-1.4

9628968E

+08

00.4

124

23.2

8Q

Q-q

gro

w7

141

302

141

302

-4.2

7987148E

+07

-4.2

7987170E

+07

00.1

521

24.7

QQ

-qisra

el

175

317

175

317

2.5

3478001E

+07

2.5

3477766E

+07

00.2

829

1.8

32.8

9Q

Q-q

pcble

nd

75

115

75

115

-7.8

4259484E

-03

-7.8

4280090E

-03

00.0

816

25.4

1


Befo

reP

reso

lvin

gA

fter

Pre

solv

ing

Pri

mal

Dual

Pre

solv

eSolv

er

Avg

Avg

Pro

pΦ

Pro

ble

mR

ow

sC

ols

Row

sC

ols

Ob

jecti

ve

Ob

jecti

ve

Tim

eT

ime

Iters

Dir

s(%

)

QQ

-qp

cb

oei1

352

727

349

724

1.1

5039007E

+07

1.1

5039019E

+07

0.0

10.3

424

1.9

65.1

8Q

Q-q

pcb

oei2

167

306

141

280

8.1

7195892E

+06

8.1

7195821E

+06

00.2

529

1.9

74.1

8Q

Q-q

pcst

air

357

615

357

615

6.2

0438137E

+06

6.2

0437660E

+06

0.0

10.3

121

23.9

9Q

Q-q

pilotn

o976

2447

955

2248

4.7

2823223E

+06

4.7

2827122E

+06

0.0

41.2

330

23.0

7Q

Q-q

recip

e92

205

84

149

-2.6

6616053E

+02

-2.6

6616081E

+02

00.0

918

22.2

7Q

Q-q

pte

st3

53

54.3

7187497E

+00

4.3

7187483E

+00

00.0

27

20

QQ

-qsc

205

206

318

204

316

-5.8

1401694E

-03

-5.8

1421806E

-03

00.1

17

22.1

3Q

Q-q

scagr2

5472

672

472

672

2.0

1737737E

+08

2.0

1737657E

+08

00.3

629

1.6

63.1

3Q

Q-q

scagr7

130

186

129

185

2.6

8659112E

+07

2.6

8658869E

+07

00.0

919

23.3

3Q

Q-q

scfx

m1

331

601

323

593

1.6

8826874E

+07

1.6

8826848E

+07

00.3

530

1.7

73.8

QQ

-qsc

fxm

2661

1201

646

1186

2.7

7761534E

+07

2.7

7761498E

+07

00.6

936

1.8

14.1

5Q

Q-q

scfx

m3

991

1801

969

1779

3.0

8163509E

+07

3.0

8163491E

+07

00.9

737

1.8

44.3

QQ

-qsc

orp

io389

467

373

456

1.8

8050938E

+03

1.8

8050937E

+03

00.1

616

22.6

5Q

Q-q

scrs

8491

1276

491

1276

9.0

4559758E

+02

9.0

4559689E

+02

00.3

824

22.9

6Q

Q-q

scsd

178

761

78

761

8.6

6666691E

+00

8.6

6666672E

+00

00.1

211

23.6

QQ

-qsc

sd6

148

1351

148

1351

5.0

8082149E

+01

5.0

8082137E

+01

00.2

15

26.9

1Q

Q-q

scsd

8398

2751

398

2751

9.4

0763587E

+02

9.4

0763560E

+02

00.3

312

24.0

3Q

Q-q

scta

p1

301

661

301

661

1.4

1586111E

+03

1.4

1586111E

+03

00.1

920

23.1

7Q

Q-q

scta

p2

1091

2501

1091

2501

1.7

3502650E

+03

1.7

3502649E

+03

00.3

813

24.7

9Q

Q-q

scta

p3

1481

3341

1481

3341

1.4

3875468E

+03

1.4

3875468E

+03

00.5

214

24.5

9Q

Q-q

seba

516

1037

516

1037

8.1

4817931E

+07

8.1

4817942E

+07

0.0

20.4

328

1.8

64.4

7Q

Q-q

share

1b

118

254

113

249

7.2

0078322E

+05

7.2

0078214E

+05

00.1

526

21.4

QQ

-qsh

are

2b

97

163

97

163

1.1

7036837E

+04

1.1

7036812E

+04

00.0

919

21.1

5Q

Q-q

shell

537

1778

537

1775

1.5

7263633E

+12

1.5

7263551E

+12

0.0

21.0

130

24.2

5Q

Q-q

ship

04l

403

2167

357

2163

2.4

2001501E

+06

2.4

2001490E

+06

00.3

215

23.2

7Q

Q-q

ship

04s

403

1507

269

1415

2.4

2499308E

+06

2.4

2499280E

+06

00.2

214

22.9

QQ

-qsh

ip08l

779

4364

695

4346

2.3

7604021E

+06

2.3

7604015E

+06

0.0

21.5

315

23.9

QQ

-qsh

ip08s

779

2468

487

2242

2.3

8572832E

+06

2.3

8572819E

+06

0.0

10.5

915

25.0

5Q

Q-q

ship

12l

1152

5534

921

5412

3.0

1887662E

+06

3.0

1887624E

+06

0.0

33.3

219

22.8

8Q

Q-q

ship

12s

1152

2870

688

2515

3.0

5696198E

+06

3.0

5696007E

+06

0.0

10.8

419

24.7

3Q

Q-q

sierr

a1228

2736

1228

2721

2.3

7504456E

+07

2.3

7504474E

+07

00.6

420

25.0

6Q

Q-q

stair

357

615

357

546

7.9

8544934E

+06

7.9

8544810E

+06

0.0

10.3

324

1.9

22.5

1Q

Q-q

standat

360

1275

359

1192

6.4

1183858E

+03

6.4

1183826E

+03

0.0

10.2

917

24.2

QQ

-s268

611

611

-2.1

2146135E

-07

-7.6

9541657E

-07

00.0

415

20

QQ

-sta

dat1

4000

6001

4000

6001

-2.8

5268640E

+07

-2.8

5268640E

+07

0.0

11.7

19

1.5

35.0

2Q

Q-s

tadat2

4000

6001

4000

6001

-3.2

6266629E

+01

-3.2

6266666E

+01

0.0

11.0

118

27.0

1Q

Q-s

tadat3

8000

12001

8000

12001

-3.5

7794523E

+01

-3.5

7794533E

+01

0.0

12.1

818

26.6

5Q

Q-t

am

e2

32

3-3

.27685749E

-10

-6.5

3153282E

-10

00.0

26

20

QQ

-ubh1

12001

18010

12001

17998

1.1

1600080E

+00

1.1

1600080E

+00

0.4

71.4

65

211.0

9Q

Q-y

ao

2001

4003

2001

4003

1.9

7704313E

+02

1.9

7704256E

+02

0.0

21.0

123

1.8

75.1

4Q

Q-z

ecevic

23

53

5-4

.12500000E

+00

-4.1

2500003E

+00

00.0

27

20

Avg

0.0

34.0

816.4

91.9

53.3

7

Tab

leA

.2:

iOptimize

per

form

an

ceon

Mit

telm

an

n’s

QC

QP

test

pro

ble

ms


Befo

reP

reso

lvin

gA

fter

Pre

solv

ing

Prim

al

Dual

Pre

solv

eSolv

er

Avg

Avg

Pro

pΦ

Pro

ble

mR

ow

sC

ols

Row

sC

ols

Ob

jectiv

eO

bje

ctiv

eT

ime

Tim

eIte

rsD

irs(%

)

airp

ort

42

84

42

84

4.7

9527017E

+04

4.7

9526915E

+04

00.1

728

1.2

91.8

6p

ola

k4

33

33

2.4

3646932E

-08

1.6

5838221E

-07

00.0

413

20

makela

12

42

4-1

.41421356E

+00

-1.4

1421356E

+00

00.0

38

24.3

5m

akela

23

33

37.2

0000270E

+00

7.2

0000353E

+00

00.0

27

25.2

6m

akela

320

21

20

21

2.3

1888348E

-11

0.0

0000000E

+00

00.0

38

20

gig

om

ez1

35

35

-2.9

9999998E

+00

-2.9

9999992E

+00

00.0

39

1.6

70

hangin

g180

288

180

288

-6.2

0176047E

+02

-6.2

0176047E

+02

00.1

610

23.4

5m

adssch

j158

81

158

81

-7.9

7283702E

+02

-7.9

7283702E

+02

00.1

411

1.6

42.2

7ro

senm

mx

45

45

-4.3

9999956E

+01

-4.3

9999945E

+01

00.0

38

20

optre

ward

28

28

-1.0

9999913E

+00

-1.0

9999913E

+00

00.0

310

20

optp

rloc

30

35

30

35

-1.6

4197690E

+01

-1.6

4197857E

+01

00.0

616

1.8

10

Avg

0.0

00.0

711.6

41.8

61.5

6

Tab

leA

.3:iOptimize

perform

an

ceof

QC

QP

onCute

testp

rob

lemp

roblem

s


Befo

reP

reso

lvin

gA

fter

Pre

solv

ing

Pri

mal

Dual

Pre

solv

eSolv

er

Avg

Avg

Pro

pΦ

Pro

ble

mR

ow

sC

ols

Row

sC

ols

Ob

jecti

ve

Ob

jecti

ve

Tim

eT

ime

Iters

Dir

s(%

)

big

bank

814

1773

814

1773

-4.2

0569612E

+06

-4.2

0569667E

+06

0.0

37.1

525

1.6

6.2

5gri

dnete

3844

7565

3844

7565

2.0

6554692E

+02

2.0

6554690E

+02

0.0

53.1

72

7.6

6gri

dnetf

3844

7565

3844

7565

2.4

2108991E

+02

2.4

2108973E

+02

0.0

427.2

315

28.1

7gri

dneth

36

61

36

61

3.9

6262686E

+01

3.9

6262685E

+01

0.0

30.2

56

20.4

3gri

dneti

36

61

36

61

4.0

2474712E

+01

4.0

2474424E

+01

0.0

30.3

98

20

dallasl

598

837

598

837

-2.0

2604132E

+05

-2.0

2610489E

+05

0.0

32.4

29

1.5

54.6

5dallasm

119

164

119

164

-4.8

1981888E

+04

-4.8

1982189E

+04

0.0

32.0

849

1.9

21.0

7dallass

29

44

29

44

-3.2

3932257E

+04

-3.2

3932858E

+04

0.0

32.0

429

1.6

90.3

pola

k1

23

23

2.7

1828182E

+00

2.7

1828183E

+00

0.0

30.2

16

20

pola

k2

211

211

5.4

5981499E

+01

5.4

5981501E

+01

0.0

30.2

16

20.5

2p

ola

k3

10

12

10

12

5.9

3300335E

+00

5.9

3300335E

+00

0.0

30.4

39

20.2

4p

ola

k5

23

23

4.9

9999999E

+01

5.0

0000001E

+01

0.0

30.2

96

20

smbank

64

117

64

117

-7.1

2929200E

+06

-7.1

2929378E

+06

0.0

31.8

530

1.3

70.6

5canti

lvr

15

15

1.3

3995635E

+00

1.3

3995641E

+00

0.0

30.6

112

20

cb2

33

33

1.9

5222455E

+00

1.9

5222408E

+00

0.0

30.2

97

20.3

6cb3

33

33

2.0

0000001E

+00

2.0

0000004E

+00

0.0

30.2

87

20

chaconn1

33

33

1.9

5222455E

+00

1.9

5222408E

+00

0.0

30.3

72

0ch

aconn2

33

33

2.0

0000001E

+00

2.0

0000004E

+00

0.0

30.2

87

20

dip

igri

47

47

6.8

0630082E

+02

6.8

0629944E

+02

0.0

30.3

99

20

gpp

498

499

498

499

1.4

4009302E

+04

1.4

4009293E

+04

0.0

32.2

810

27.7

hong

14

14

1.3

4730701E

+00

1.3

4730660E

+00

0.0

30.6

616

20.3

1lo

adbal

31

51

31

51

4.5

2851044E

-01

4.5

2850995E

-01

0.0

30.8

121

20.2

5sv

anb

erg

5000

5000

5000

5000

8.3

6142276E

+03

8.3

6142275E

+03

0.1

1114.3

316

211.2

4ante

nna-a

nte

nna2

166

49

166

49

1.5

7986028E

+00

1.5

7986028E

+00

0.0

37.5

315

28.3

ante

nna-a

nte

nna

vare

ps

166

50

166

50

1.2

2309383E

+00

1.2

2309383E

+00

0.0

312.1

218

1.7

26.9

5bra

ess

-tra

fequil

361

722

361

722

5.2

6883657E

+01

5.2

6883559E

+01

0.0

31.2

517

21.4

6bra

ess

-tra

fequil2

437

798

437

798

5.2

6883639E

+01

5.2

6883616E

+01

0.0

31.8

614

27.7

3bra

ess

-tra

fequilsf

309

856

309

856

5.1

5910455E

+01

5.1

5910406E

+01

0.0

21.8

117

20.7

6bra

ess

-tra

fequil2sf

385

932

385

932

5.1

5910445E

+01

5.1

5910425E

+01

0.0

32.3

313

27.5

8ele

na-c

hem

eq

12

38

12

38

-1.9

1087024E

+03

-1.9

1087054E

+03

0.0

10.9

519

1.6

30.2

1ele

na-s

383

114

114

7.2

8593644E

+05

7.2

8593346E

+05

0.0

31.2

819

1.4

20.0

8firfi

lter-

fir

convex

243

163

243

163

1.0

4648765E

+00

1.0

4648766E

+00

0.0

30.8

314

26.9

9m

ark

ow

itz-g

row

thopt

18

18

-1.1

7084823E

-01

-1.1

7084822E

-01

0.0

40.3

69

20

wbv-a

nte

nna2

1197

1180

1197

1180

1.0

9359976E

+00

1.0

9359978E

+00

0.0

185.4

614

29.1

6w

bv-l

ow

pass

2200

219

200

219

1.0

5905136E

+00

1.0

5905055E

+00

08.3

225

26.9

9batc

h73

106

73

106

2.5

9180347E

+05

2.5

9175929E

+05

0.0

10.3

813

21.3

7st

ock

cycle

97

481

97

481

1.1

7916213E

+05

1.1

7915981E

+05

00.3

611

21.4

4sy

nth

es1

610

610

7.5

9284403E

-01

7.5

9284359E

-01

00.1

97

20.5

3sy

nth

es2

14

21

14

21

-5.5

4416573E

-01

-5.5

4416857E

-01

00.2

18

20

synth

es3

23

34

23

34

1.5

0821846E

+01

1.5

0821817E

+01

0.0

10.2

18

20

trim

loss

224

53

24

53

7.1

8306638E

-01

7.1

8306802E

-01

0.0

20.2

611

20.7

9tr

imlo

ss4

64

145

64

145

1.7

0933108E

+00

1.7

0933115E

+00

0.0

10.2

811

22.5

1tr

imlo

ss5

90

216

90

216

1.1

7887002E

+00

1.1

7887282E

+00

00.3

713

24.0

2tr

imlo

ss6

120

287

120

287

1.3

0565671E

+00

1.3

0565766E

+00

0.0

10.3

513

25.1

9tr

imlo

ss7

154

436

154

436

5.9

3497411E

-01

5.9

3498455E

-01

00.6

615

27.8

3tr

imlo

ss12

384

1028

384

1028

2.3

1187222E

+00

2.3

1187205E

+00

06.8

118

28.8

5

Avg

0.0

26.5

514.0

91.9

42.9

4

Tab

leA

.4:

iOptimize

per

form

an

ceonCute

CP

test

pro

ble

ms


Pro

ble

mMOSEK

iOptimize

Pro

ble

mMOSEK

iOptimize

Pro

ble

mMOSEK

iOptimize

aug2d

612

lase

r12

10

qp

cb

oei1

20

22

aug2dc

612

liswet1

26

19

qp

cb

oei2

21

26

aug2dcqp

13

13

liswet2

32

17

qp

csta

ir22

25

aug2dqp

15

14

liswet3

21

16

qpilo

tno

35§

42

aug3d

68

liswet4

26§

18

qre

cip

e16

19

aug3dc

68

liswet5

25

16

qpte

st6

6aug3dcqp

11

11

liswet6

24

18

qsc

205

16

17

aug3dqp

12

13

liswet7

31

19

qsc

agr2

520

22

cont-0

50

11

11

liswet8

38

37

qsc

agr7

19

22

cont-1

00

12

11

liswet9

37

31

qsc

fxm

125

25

cont-1

01

13

9lisw

et1

036

21

qsc

fxm

230

33

cont-2

00

12

11

liswet1

131

22

qsc

fxm

330

35

cont-2

01

15

10

liswet1

230

32

qsc

orp

io13

14

cont-3

00

16

13

lotsch

d10

9qsc

rs821

24

cvxqp1

m10

10

mosa

rqp1

99

qsc

sd1

10

9cvxqp1

s8

8m

osa

rqp2

99

qsc

sd6

13

14

cvxqp2

m10

10

pow

ell2

010§

8qsc

sd8

12

11

cvxqp2

s9

10

prim

al1

10

9qsc

tap1

18

17

cvxqp3

m12

12

prim

al2

87

qsc

tap2

13

13

cvxqp3

s10

10

prim

al3

99

qsc

tap3

14

15

dpklo

15

3prim

al4

10

9qse

ba

16

26

dto

c3

54

prim

alc

118

18

qsh

are

1b

23

29

dual1

14

12

prim

alc

215

25

qsh

are

2b

17

17

dual2

12

9prim

alc

58

10

qsh

ell

33

38

dual3

12

9prim

alc

810

12

qsh

ip04l

14

14

dual4

12

10

q25fv

47

28

27

qsh

ip04s

13

14

dualc

110

25

qadlittle

11

11

qsh

ip08l

13

14

dualc

210

20

qafiro

12

14

qsh

ip08s

13

14

dualc

57

13

qbandm

19

20

qsh

ip12l

17

16

dualc

87

19

qb

eaconf

14

14

qsh

ip12s

17

17

genhs2

84

3qb

ore

3d

13

19

qsie

rra18

20

gould

qp2

14

13

qbra

ndy

14

17

qsta

ir22

23

gould

qp3

10

11

qcapri

26

36

qsta

ndat

15

13

hs1

18

10

10

qe226

14

16

s268

22

12

hs2

19

9qeta

macr

27

32

stadat1

25

12

hs2

68

22

12

qff

fff8

022

25

stadat2

41

18

hs3

55

5qfo

rpla

n23

31

stadat3

46

19

hs3

5m

od

11

11

qgfrd

xpn

23

30

tam

e3

3hs5

15

3qgro

w15

22

22

ubh1

63

hs5

24

3qgro

w22

25

26

yao

23

23

hs5

36

4qgro

w7

23

21

zecevic

25

5hs7

65

5qisra

el

21

28

ksip

21

16

qp

cble

nd

19

17

Avg

16.2

215.9

2

1§:

Near

optim

al

solu

tion

found.

Tab

leA

.5:C

omp

arisonw

ithMOSEK

on

Maro

san

dM

eszaro

s’sQ

Ptest

prob

lems


Pro

ble

mMOSEK

iOptimize

Pro

ble

mMOSEK

iOptimize

Pro

ble

mMOSEK

iOptimize

QQ

-aug2d

11

8Q

Q-l

ase

r135

19

QQ

-qp

cb

oei1

24

24

QQ

-aug2dc

98

QQ

-lis

wet1

41?

21

QQ

-qp

cb

oei2

29

29

QQ

-aug2dcqp

18

19

QQ

-lis

wet2

21

17

QQ

-qp

cst

air

24

21

QQ

-aug2dqp

21

QQ

-lis

wet3

17

17

QQ

-qpilotn

o39

30

QQ

-aug3d

65

QQ

-lis

wet4

17

18

QQ

-qre

cip

e16

18

QQ

-aug3dc

75

QQ

-lis

wet5

15

20

QQ

-qpte

st7

7Q

Q-a

ug3dcqp

17

11

QQ

-lis

wet6

17

19

QQ

-qsc

205

17

17

QQ

-aug3dqp

22

12

QQ

-lis

wet7

34?

22

QQ

-qsc

agr2

520

29

QQ

-cont-

050

14

11

QQ

-lis

wet8

49?

30

QQ

-qsc

agr7

18

19

QQ

-cont-

100

15

12

QQ

-lis

wet9

26?

36

QQ

-qsc

fxm

143

30

QQ

-cont-

101

12

11

QQ

-lis

wet1

018?

29

QQ

-qsc

fxm

239

36

QQ

-cont-

200

19

13

QQ

-lis

wet1

127?

23

QQ

-qsc

fxm

326

37

QQ

-cont-

201

14

12

QQ

-lis

wet1

224?

38

QQ

-qsc

orp

io17

16

QQ

-cont-

300

15

12

QQ

-lots

chd

15

9Q

Q-q

scrs

824

24

QQ

-cvxqp1

m15

10

QQ

-mosa

rqp1

15

10

QQ

-qsc

sd1

13

11

QQ

-cvxqp1

s15

8Q

Q-m

osa

rqp2

14

9Q

Q-q

scsd

617

15

QQ

-cvxqp2

m18

10

QQ

-pow

ell20

19

QQ

-qsc

sd8

16

12

QQ

-cvxqp2

s12

9Q

Q-p

rim

al1

11

10

QQ

-qsc

tap1

19

20

QQ

-cvxqp3

m18

13

QQ

-pri

mal2

99

QQ

-qsc

tap2

18

13

QQ

-cvxqp3

s14

9Q

Q-p

rim

al3

10

11

QQ

-qsc

tap3

18

14

QQ

-dpklo

16

5Q

Q-p

rim

al4

11

9Q

Q-q

seba

62

28

QQ

-dto

c3

11

7Q

Q-p

rim

alc

120

19

QQ

-qsh

are

1b

27

26

QQ

-dual1

14

13

QQ

-pri

malc

218

17

QQ

-qsh

are

2b

19

19

QQ

-dual2

13

12

QQ

-pri

malc

511

10

QQ

-qsh

ell

26

30

QQ

-dual3

14

12

QQ

-pri

malc

815

12

QQ

-qsh

ip04l

21

15

QQ

-dual4

13

10

QQ

-q25fv

47

39

28

QQ

-qsh

ip04s

18

14

QQ

-dualc

117

21

QQ

-qadlitt

le19

12

QQ

-qsh

ip08l

20

15

QQ

-dualc

214

17

QQ

-qafiro

16

14

QQ

-qsh

ip08s

21

15

QQ

-dualc

511

9Q

Q-q

bandm

21

21

QQ

-qsh

ip12l

24

19

QQ

-dualc

818

13

QQ

-qb

eaconf

12

13

QQ

-qsh

ip12s

24

19

QQ

-genhs2

85

5Q

Q-q

bore

3d

14

19

QQ

-qsi

err

a24

20

QQ

-gould

qp2

13

15

QQ

-qbra

ndy

17

17

QQ

-qst

air

31

24

QQ

-gould

qp3

13

10

QQ

-qcapri

26

32

QQ

-qst

andat

16

17

QQ

-hs1

18

11

11

QQ

-qe226

17

17

QQ

-s268

13

15

QQ

-hs2

113

10

QQ

-qeta

macr

24

33

QQ

-sta

dat1

53

19

QQ

-hs2

68

13

15

QQ

-qff

fff8

030

26

QQ

-sta

dat2

18

18

QQ

-hs3

56

6Q

Q-q

forp

lan

180§

23

QQ

-sta

dat3

18

18

QQ

-hs3

5m

od

13

12

QQ

-qgfr

dxpn

25

24

QQ

-tam

e5

6Q

Q-h

s51

55

QQ

-qgro

w15

21

21

QQ

-ubh1

20

5Q

Q-h

s52

65

QQ

-qgro

w22

24

24

QQ

-yao

27§

23

QQ

-hs5

310

6Q

Q-q

gro

w7

21

21

QQ

-zecevic

26

7Q

Q-h

s76

86

QQ

-qis

rael

28

29

Avg1

20.9

116.4

3Q

Q-k

sip

15

13

QQ

-qp

cble

nd

20

16

Avg2

18.3

416.2

6

1:

Max

num

ber

of

itera

tions

reach

ed.

2§:

Near

opti

mal

solu

tion

found.

3?:

Term

inate

dw

ith

unknow

nst

atu

s.

Tab

leA

.6:

Com

par

ison

wit

hMOSEK

on

Mit

telm

an

n’s

QC

QP

test

pro

ble

ms


Problem MOSEK iOptimize

airport 26 28polak4 10 13makela1 8 8makela2 7 7makela3 6 8gigomez1 8 9hanging 13 10madsschj 10 11rosenmmx 9 8optreward 10 10optprloc 18 16

Avg 11.36 11.64

Table A.7: Comparison with MOSEK on Cute QCQP test problems


Problem MOSEK iOptimize Ipopt Knitro

bigbank 20 21 24 19gridnete 9 7 5 4gridnetf 16 13 27 17gridneth 8 6 6 3gridneti 10 8 7 6dallasl 74 33 249 245dallasm 128 31

dallass 31 135polak1 6 5 4 2polak2 6 5 4 1polak3 10 8

polak5 6 5 24 12smbank 14 27 12 11cantilvr 12 12 5 3cb2 8 8 4 3cb3 7 7 4 4chaconn1 8 8 4 3chaconn2 7 7 4 4dipigri 14 10 6 2gpp 12 10 22 11hong 13 15 11 7loadbal 22 21 13 7svanberg 14 14 30 16

antenna\antenna2 21§ 16 178 58antenna\antenna vareps 19 18 106 61braess\trafequil 15 16 18 22braess\trafequil2 11 13 26 13braess\trafequilsf 14 15 18 24braess\trafequil2sf 12 12 25 12elena\chemeq 23 20 37 16elena\s383 85 21 18 8firfilter\fir convex 33§ 13 27 11markowitz\growthopt 8 7 8 7wbv\antenna2 15 13 31 16wbv\lowpass2 30§ 22 31 66

batch 13 12 13 16stockcycle 11 10 23 12synthes1 6 6 6 6synthes2 8 8 10 7synthes3 8 8 9 8trimloss2 10 10 13 7trimloss4 11 12 16 8trimloss5 13 13 20 10trimloss6 12 12 23 11trimloss7 13 13 35 14trimloss12 19 17 41 22

Avg1 18.46 14.00 – –Avg2 13.07 12.69 – –

1 : Max number of iterations reached.2 §: Near optimal solution found.3 : Function evaluate error from AMPL.

Table A.8: Comparison with MOSEK, Ipopt, and Knitro on CP test problems


Presolve Solver Avg Avg Prop ΦProblem Time Time Iters Dirs (%)

iQQ-aug2d 0.01 7.14 16 1.69 4.78iQQ-aug2dc 0.01 6.8 16 1.69 4.34iQQ-aug2dcqp 0.78 13.44 29 1.34 4.5iQQ-aug2dqp 0.73 13.4 29 1.34 4.44iQQ-aug3d 0 4.72 35 1.09 2.43iQQ-aug3dc 0 1.06 16 1.5 5.13iQQ-aug3dcqp 0.04 2.59 33 1.09 4.1iQQ-aug3dqp 0.03 2.68 37 1.05 4.4iQQ-cont-050 0 0.76 9 2 4.72iQQ-cont-100 0.01 6.7 14 1.93 2.17iQQ-cont-101 0 3.33 9 2 2.35iQQ-cont-200 0.03 47.74 14 2 1.71iQQ-cont-201 0.03 26.08 9 2 2.09iQQ-cont-300 0.05 104.79 9 2 1.57iQQ-cvxqp1 m 0 2.19 26 1.38 1.38iQQ-cvxqp1 s 0 2.24 35 1.2 0.45iQQ-cvxqp2 m 0 2.14 29 1.34 0.9iQQ-cvxqp2 s 0 2.41 37 1.19 0.92iQQ-cvxqp3 m 0 2.85 30 1.23 1.8iQQ-cvxqp3 s 0 1.22 35 1.17 0.98iQQ-dpklo1 0 0.36 13 2 0.83iQQ-dtoc3 0.32 4.21 15 1.87 5.9iQQ-dual1 0 1.14 31 1.32 0.88iQQ-dual2 0 0.52 16 1.69 0iQQ-dual3 0 0.63 18 1.67 1.61iQQ-dual4 0 1.23 33 1.33 0iQQ-dualc1 0 1.06 35 1.66 0iQQ-dualc2 0 1.19 35 1.57 0.59iQQ-dualc5 0 1.65 44 1.32 1.04iQQ-dualc8 0 1.06 34 1.68 1.9iQQ-genhs28 0 0.37 13 2 0iQQ-gouldqp2 0 0.27 10 2 0iQQ-gouldqp3 0 0.56 16 1.63 1.82iQQ-hs118 0 0.27 11 2 0iQQ-hs21 0.02 1.2 35 1.11 0.08iQQ-hs268 0 1.52 45 1.11 0iQQ-hs35 0 0.52 16 1.75 0iQQ-hs35mod 0 0.54 16 1.75 0.19iQQ-hs51 0 1.07 27 1.19 0iQQ-hs52 0 0.73 26 1.31 0iQQ-hs53 0 1.41 39 1.13 0iQQ-hs76 0 0.36 13 2 0iQQ-ksip 0 1.73 42 1.05 3.3



iQQ-laser 0 0.4 13 2 5.22iQQ-liswet1 0.01 16.43 42 1.14 4.19iQQ-liswet2 0.02 17.23 43 1.12 4.9iQQ-liswet3 0.01 15 38 1.16 5.15iQQ-liswet4 0.01 15.72 40 1.13 5.16iQQ-liswet5 0.01 18.09 47 1.17 4.49iQQ-liswet6 0.02 16.79 43 1.14 4.64iQQ-liswet7 0.01 15.81 40 1.13 5.12iQQ-liswet8 0.01 14.03 33 1.15 4.37iQQ-liswet9 0.01 15.69 37 1.14 4.69iQQ-liswet10 0.02 16.5 42 1.14 4.97iQQ-liswet11 0.01 16.33 42 1.14 4.9iQQ-liswet12 0.01 15.73 38 1.13 4.62iQQ-lotschd 0 0.3 12 2 0iQQ-mosarqp1 0 2.04 33 1.09 4.55iQQ-mosarqp2 0 1.41 38 1.08 3.57iQQ-powell20 0.02 6.5 16 1.63 3.5iQQ-primal1 0.01 1.28 43 1.09 1.41iQQ-primal2 0 1.31 37 1.14 2.47iQQ-primal3 0 1.58 41 1.1 2.24iQQ-primal4 0.01 1.56 36 1.17 4.44iQQ-primalc1 0 1.37 42 1.26 1.03iQQ-primalc2 0 1.32 36 1.33 1.52iQQ-primalc5 0 0.97 23 1.39 0.93iQQ-primalc8 0 1.1 32 1.28 1.01iQQ-q25fv47 0.01 1.4 15 2 6.42iQQ-qadlittle 0 0.34 14 2 0.59iQQ-qafiro 0 1.37 39 1.15 0.07iQQ-qbandm 0 0.34 14 1.86 3.02iQQ-qbeaconf 0 0.41 14 1.93 0.5iQQ-qbore3d 0 0.51 18 1.89 0iQQ-qbrandy 0 0.48 16 1.88 0.41iQQ-qcapri 0 0.62 22 1.91 1.63iQQ-qe226 0 1.41 36 1.22 1.85iQQ-qetamacr 0 1.09 34 1.21 3.17iQQ-qfffff80 0 0.57 20 1.8 3.57iQQ-qforplan 0.01 0.25 11 2 1.63iQQ-qgfrdxpn 0.01 0.47 19 2 2.42iQQ-qgrow15 0 0.44 16 1.94 1.15iQQ-qgrow22 0.01 0.54 18 1.89 0iQQ-qgrow7 0 0.56 18 1.89 0.54iQQ-qisrael 0 0.47 19 2 1.3iQQ-qpcblend 0 1.5 38 1.16 0.27



iQQ-qpcboei1 0.01 0.31 12 2 6.06iQQ-qpcboei2 0 0.44 17 1.94 4.65iQQ-qpcstair 0.01 0.64 20 1.75 1.27iQQ-qpilotno 0.04 0.65 16 2 3.84iQQ-qrecipe 0 0.75 20 1.6 0.13iQQ-qptest 0 0.56 16 1.81 0.18iQQ-qsc205 0 1.51 41 1.24 1.35iQQ-qscagr25 0 0.32 14 2 0.32iQQ-qscagr7 0 0.33 14 2 1.22iQQ-qscfxm1 0 0.4 15 1.93 1.79iQQ-qscfxm2 0 0.42 16 1.94 2.44iQQ-qscfxm3 0 0.54 16 1.94 4.71iQQ-qscorpio 0 0.95 25 1.24 1.91iQQ-qscrs8 0 0.63 22 1.73 6.45iQQ-qscsd1 0 0.84 20 1.6 1.31iQQ-qscsd6 0 0.89 22 1.64 2.07iQQ-qscsd8 0 2.02 37 1.11 5.11iQQ-qsctap1 0 1.07 36 1.19 1.14iQQ-qsctap2 0 2.26 40 1.13 3.58iQQ-qsctap3 0 2.26 32 1.22 3.65iQQ-qseba 0.02 0.57 20 1.85 5.36iQQ-qshare1b 0 0.44 17 1.94 2.33iQQ-qshare2b 0 0.47 13 1.69 2.13iQQ-qshell 0.02 0.82 17 2 6.4iQQ-qship04l 0 0.31 11 2 0iQQ-qship04s 0.01 0.25 11 2 6.88iQQ-qship08l 0.01 1.63 16 1.94 4.56iQQ-qship08s 0 0.72 17 1.88 3.73iQQ-qship12l 0.01 2.96 17 1.71 4.74iQQ-qship12s 0.01 0.99 18 1.83 2.98iQQ-qsierra 0 0.64 17 2 5iQQ-qstair 0 0.39 17 2 1.05iQQ-qstandat 0.01 1.37 39 1.18 2.05iQQ-s268 0 1.56 45 1.11 0.06iQQ-stadat1 0.01 3.13 26 1.23 4.19iQQ-stadat2 0 4.39 35 1.14 4.42iQQ-stadat3 0.01 6.18 25 1.2 4.25iQQ-tame 0 0.23 10 2 0iQQ-ubh1 0.46 5.73 26 1.96 6.21iQQ-yao 0.01 3.17 38 1.05 4.85iQQ-zecevic2 0 0.23 9 2 0

Avg 0.02 4.40 25.30 1.56 2.48

Table A.9: iOptimize performance on Maros and Meszaros’s QCQP infeasible testproblems


iairport 0.01 0.6 19 1.63 0.67ipolak4 0 0.81 25 1.28 0.12imakela1 0 1.11 32 1.13 0imakela2 0 1.13 32 1.13 0imakela3 0 0.32 14 2 0igigomez1 0 1.02 26 1.19 0ihanging 0 1.17 30 1.13 4.27imadsschj 0.06 0.53 17 1.59 0.76irosenmmx 0 1 35 1.14 0ioptreward 0 0.64 20 1.55 0ioptprloc 0 0.92 21 1.52 0.11

Avg 0.01 0.84 24.64 1.39 0.54

Table A.10: iOptimize performance on Cute QCQP infeasible test problems



ibigbank 0 17.12 27 1.33 6.06igridnete 0 81.51 14 1.71 6.79igridnetf 0 377.58 36 1.08 5.67igridneth 0.02 0.78 23 1.39 0.26igridneti 0 1.23 40 1.07 1.15idallaslidallasm 0.01 1.96 47 1.4 2.26idallass 0 1.58 45 1.22 0.51ipolak1 0 0.41 16 1.81 0ipolak2 0 0.32 13 2 0ipolak3 0 0.77 27 1.3 0.52ipolak5 0 0.41 13 2 0.25ismbank 0 1.13 31 1.13 1.7icantilvr 0 0.29 12 2 0.35icb2 0 0.94 32 1.19 0icb3 0.01 1.25 31 1.16 0.24ichaconn1 0 0.95 32 1.19 0ichaconn2 0 1.26 31 1.16 0.08idipigri 0 1.29 32 1.19 0.08igpp 0 34.24 38 1.11 5.34ihong 0 0.92 22 1.45 0.11iloadbal 0 0.37 14 2 0.27isvanberg 0.08 632.99 27 1.22 6.65

antenna-iantenna2 0 68.79 186 1.65 13.31antenna-iantenna vareps 0 48.02 47 1.13 5.83braess-itrafequil 0 4.77 38 1.37 3.1braess-itrafequil2 0 1.87 9 2 7.34braess-itrafequilsf 0 9.23 46 1.26 2.71braess-itrafequil2sf 0 2.56 9 2 7.16elena-ichemeq 0 0.06 14 2 4.92elena-is383 0 0.06 14 1.93 1.64firfilter-ifir convex 0 0.75 13 1.92 6.03markowitz-igrowthopt 0 0.08 16 1.69 0wbv-iantenna2 0.06 119.88 17 2 7.97wbv-ilowpass2 0 4.54 13 1.92 7.35

ibatch 0 0.15 22 1.36 3.47istockcycle 0 0.88 35 1.14 6.29isynthes1 0 0.09 25 1.36 0isynthes2 0 0.09 23 1.35 1.15isynthes3 0 0.18 40 1.1 3.91itrimloss2 0 0.1 15 1.93 0itrimloss4 0 0.15 16 1.94 7.43itrimloss5 0 0.22 15 2 6.07itrimloss6 0 0.37 15 2 7.32itrimloss7 0 0.84 16 2 6.58itrimloss12 0.01 7.79 16 2 7.41

Avg 0.00 31.79 28.07 1.56 3.45

1 : Function evaluate error from AMPL.

Table A.11: iOptimize performance on Cute CP infeasible test problems


Pro

ble

mMOSEK

iOptimize

Pro

ble

mMOSEK

iOptimize

Pro

ble

mMOSEK

iOptimize

iQQ

-aug2d

16

iQQ

-lase

r29

13

iQQ

-qp

cb

oei1

12

iQQ

-aug2dc

16

iQQ

-liswet1

31

42

iQQ

-qp

cb

oei2

17

iQQ

-aug2dcqp

29

iQQ

-liswet2

30

43

iQQ

-qp

csta

ir

20

iQQ

-aug2dqp

24

29

iQQ

-liswet3

30

38

iQQ

-qpilo

tno

16

iQQ

-aug3d

21

35

iQQ

-liswet4

30

40

iQQ

-qre

cip

e12

20

iQQ

-aug3dc

16

16

iQQ

-liswet5

21

47

iQQ

-qpte

st15

16

iQQ

-aug3dcqp

121

33

iQQ

-liswet6

30

43

iQQ

-qsc

205

16

41

iQQ

-aug3dqp

95

37

iQQ

-liswet7

24

40

iQQ

-qsc

agr2

5383?

14

iQQ

-cont-0

50

9

iQQ

-liswet8

26

33

iQQ

-qsc

agr7

38

14

iQQ

-cont-1

00

14

iQQ

-liswet9

24

37

iQQ

-qsc

fxm

1

15

iQQ

-cont-1

01

9

iQQ

-liswet1

030

42

iQQ

-qsc

fxm

2

16

iQQ

-cont-2

00

14

iQQ

-liswet1

130

42

iQQ

-qsc

fxm

3

16

iQQ

-cont-2

01

70

9iQ

Q-lisw

et1

224

38

iQQ

-qsc

orp

io31

25

iQQ

-cont-3

00

9

iQQ

-lotsch

d26

12

iQQ

-qsc

rs852

22

iQQ

-cvxqp1

m

26

iQQ

-mosa

rqp1

100

33

iQQ

-qsc

sd1

28

20

iQQ

-cvxqp1

s21

35

iQQ

-mosa

rqp2

66

38

iQQ

-qsc

sd6

25

22

iQQ

-cvxqp2

m97

29

iQQ

-pow

ell2

0

16

iQQ

-qsc

sd8

30

37

iQQ

-cvxqp2

s18

37

iQQ

-prim

al1

31

43

iQQ

-qsc

tap1

38

36

iQQ

-cvxqp3

m

30

iQQ

-prim

al2

32

37

iQQ

-qsc

tap2

40

iQQ

-cvxqp3

s22

35

iQQ

-prim

al3

33

41

iQQ

-qsc

tap3

87

32

iQQ

-dpklo

1

13

iQQ

-prim

al4

34

36

iQQ

-qse

ba

28

20

iQQ

-dto

c3

24

15

iQQ

-prim

alc

136

42

iQQ

-qsh

are

1b

51

17

iQQ

-dual1

10

31

iQQ

-prim

alc

219

36

iQQ

-qsh

are

2b

175

13

iQQ

-dual2

916

iQQ

-prim

alc

522

23

iQQ

-qsh

ell

20

17

iQQ

-dual3

918

iQQ

-prim

alc

824

32

iQQ

-qsh

ip04l

15

11

iQQ

-dual4

933

iQQ

-q25fv

47

15

iQQ

-qsh

ip04s

15

11

iQQ

-dualc

118

35

iQQ

-qadlittle

14

iQQ

-qsh

ip08l

16

iQQ

-dualc

215

35

iQQ

-qafiro

70

39

iQQ

-qsh

ip08s

17

iQQ

-dualc

510

44

iQQ

-qbandm

37

14

iQQ

-qsh

ip12l

17

iQQ

-dualc

830

34

iQQ

-qb

eaconf

26

14

iQQ

-qsh

ip12s

18

iQQ

-genhs2

8

13

iQQ

-qb

ore

3d

10

18

iQQ

-qsie

rra

17

iQQ

-gould

qp2

10

10

iQQ

-qbra

ndy

16

iQQ

-qsta

ir353?

17

iQQ

-gould

qp3

14

16

iQQ

-qcapri

22

iQQ

-qsta

ndat

340

39

iQQ

-hs1

18

232

11

iQQ

-qe226

281

36

iQQ

-s268

26

45

iQQ

-hs2

140

35

iQQ

-qeta

macr

34

iQQ

-stadat1

16

26

iQQ

-hs2

68

26

45

iQQ

-qff

fff8

0

20

iQQ

-stadat2

35

iQQ

-hs3

572

16

iQQ

-qfo

rpla

n21

11

iQQ

-stadat3

25

iQQ

-hs3

5m

od

25

16

iQQ

-qgfrd

xpn

21

19

iQQ

-tam

e6

10

iQQ

-hs5

129

27

iQQ

-qgro

w15

22

16

iQQ

-ubh1

26

iQQ

-hs5

222

26

iQQ

-qgro

w22

25

18

iQQ

-yao

27

38

iQQ

-hs5

327

39

iQQ

-qgro

w7

22

18

iQQ

-zecevic

28

9iQ

Q-h

s76

69

13

iQQ

-qisra

el

19

Avg1

42.2

028.1

5iQ

Q-k

sip30

42

iQQ

-qp

cble

nd

105

38

Acg2

27.3

928.1

5

1:

Max

num

ber

of

itera

tions

reach

ed.

2?:

Term

inate

dw

ithunknow

nsta

tus.

Tab

leA

.12:

Com

parison

with

MOSEK

on

Maro

san

dM

eszaro

s’sQ

CQ

Pin

feasible

testp

roblem

s


Problem MOSEK iOptimize

iairport 10 19ipolak4 25imakela1 28 32imakela2 29 32imakela3 8 14igigomez1 26 26ihanging 32 30imadsschj 24 17irosenmmx 27 35ioptreward 11 20ioptprloc 8 21

Avg 20.30 24.60

1 : Max number of iterationsreached.

Table A.13: Comparison with MOSEK on Cute QCQP infeasible test problems


Problem MOSEK iOptimize Ipopt Knitro

ibigbank 196 27 48 42igridnete 134 14 21

igridnetf 36 42

igridneth 202 23 25 18igridneti 40 40 133§idallasl 50§

idallasm 47 0idallass 45 0ipolak1 50 16

ipolak2 7 13

ipolak3 178 27

ipolak5 7 13 37

ismbank 165 31 34 37icantilvr 0 12 25 0icb2 196 32 18icb3 196 31 8ichaconn1 196 32 6ichaconn2 196 31 10 17idipigri 157 32 188

igpp 38 65

ihong 11 22 16

iloadbal 123 14 22

isvanberg 27 48 53

antenna\iantenna2 186 156antenna\iantenna vareps 47

braess\itrafequil 43 38 36 126braess\itrafequil2 7 9 82 50braess\itrafequilsf 47 46 34 103braess\itrafequil2sf 9 54 91elena\ichemeq 14 14 102 16elena\is383 7 14 67

firfilter\ifir convex 0 13 99 0markowitz\igrowthopt 15 16 22

wbv\iantenna2 0 17 32 0wbv\ilowpass2 0 13 32 0

ibatch 0 22 32 158istockcycle 117 35 72

isynthes1 9 25 35 35isynthes2 62 23 48 46isynthes3 53 40 94

itrimloss2 0 15 52 0itrimloss4 0 16 40 0itrimloss5 0 15 60 0itrimloss6 0 15 55 0itrimloss7 0 16 61 0itrimloss12 0 16 73 0

Avg1 66.33 19.44 – –Avg2 11.74 16.04 – –

1 : Max number of iterations reached.2 : Function evaluate error from AMPL.3 §: Solver terminates with a near optimal status.4 : Ipopt fails in restoration phase.

Table A.14: Comparison with MOSEK, Ipopt, and Knitro on CP infeasible test prob-lems

SOLUTION OF MONOTONE COMPLEMENTARITY AND GENERAL CONVEX PROGRAMMING...

Documents

Transcript of SOLUTION OF MONOTONE COMPLEMENTARITY AND GENERAL CONVEX PROGRAMMING...