Structure of Valid Inequalities for Mixed Integer Conic Programs · 2014. 1. 6. · Mixed Integer...

Structure of Valid Inequalities for Mixed Integer ConicPrograms

Fatma Kılınc-Karzan

Tepper School of BusinessCarnegie Mellon University

18th Combinatorial Optimization WorkshopAussois, France

January 6-10, 2014

F. Kılınc-Karzan (CMU) Structure of Valid Inequalities for MICPs 1 / 34

Outline

Mixed integer conic optimization

MotivationProblem setting

Structure of linear valid inequalities

K-minimal valid inequalitiesK-sublinear valid inequalities

Necessary conditionsSufficient conditions

Disjunctive cuts for Lorentz cone (joint work with Sercan Yıldız)

Connection to the new frameworkDeriving the nonlinear valid inequality

Mixed Integer Conic Programming

Mixed Integer Linear Program

min cT x

s.t. Ax ≥ b

x ∈ Zd × Rn−d

min cT x

s.t. Ax − b ∈ Rm+

x ∈ Zd × Rn−d

min cT

Mixed Integer Convex Program

min cT x

s.t. x ∈ Q

x ∈ Zd × Rn−d

where Q is a closed convex setmin cT

min cT x

x ∈ Zd × Rn−d

min cT

min cT x

s.t. x ∈ Q

x ∈ Zd × Rn−d

min cT x

x ∈ Zd × Rn−d

min cT

min cT x

s.t. x ∈ Q

x ∈ Zd × Rn−d

min cT x

x ∈ Zd × Rn−d

min cT

Mixed Integer Conic Program

min cT x

s.t. Ax − b ∈ Kx ∈ Zd × Rn−d

where K is a convex cone

min cT x

x ∈ Zd × Rn−d

min cT

min cT x

Nonnegative orthantRm

+ = {y ∈ Rm : yi ≥ 0 ∀i}

Lorentz coneLm = {y ∈ Rm : ym ≥

2 + . . . + y2m−1}

Positive semidefinite coneSm

+ = {X ∈ Rm×m : aTXa ≥ 0 ∀a ∈ Rm}

min cT x

x ∈ Zd × Rn−d

min cT

min cT x

min cT

Motivation

A good understanding of mixed integer linear programs (MIPs)

⇒ Well known classes of valid inequalities, closures, ...

[C-G cuts, MIR inequalities, Split cuts, Disjunctive cuts, ...]

⇒ Characterization of minimal and extremal inequalities for linear MIPs

Recent and growing interest in mixed integer conic programs (MICPs)

⇒ Development of general classes of valid inequalities for conic sets

[Extensions of C-G cuts, MIR inequalities, Split cuts, Intersection cuts,

Disjunctive cuts, ...]

⇒ Recent results on C-G closures for convex sets, conic MIP duality

Motivation

Success of solvers depend on identifying and efficiently separatingstrong valid inequalities

Is it possible to develop a framework to establish strength of valid linearinequalities for MICPs?

Problem Setting

We are interested in the convex hull of the following set:

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}where

E is a finite dimensional Euclidean space with inner product 〈·, ·〉A is a linear map from E to Rm

B ⊂ Rm is a given set of points (can be finite or infinite)

K ⊂ E is a full-dimensional, closed, convex and pointed cone

Nonnegative orthant, Rn+ := {x ∈ Rn : xi ≥ 0 ∀i}

Lorentz cone, Ln := {x ∈ Rn : xn ≥√x2

2 + . . .+ x2n−1 }

Positive semidefinite cone, Sn+ := {X ∈ Rn×n : aTXa ≥ 0 ∀a ∈ Rn}

Representation flexibility

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

This set captures the essential structure of MICPs (can also be used as a

natural relaxation):

{(y , v) ∈ Rk+ × Zq : Wy + Hv − b ∈ K}

where K ⊂ E is a full-dimensional, closed, convex, pointed cone.Define

), A =

[W , −Id

], and B = b − HZq,

where Id is the identity map in E . Then we arrive at

S(A,K′,B) = {x ∈ (Rk × E ) : Ax ∈ B, x ∈ (Rk+ ×K)︸︷︷︸:=K′

Many more examples involving modeling disjunctions, complementarityrelations, Gomory’s Corner Polyhedron (1969), etc...

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

{(y , v) ∈ Rk+ × Zq : Wy + Hv − b ∈ K}

where K ⊂ E is a full-dimensional, closed, convex, pointed cone.

Define

), A =

[W , −Id

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

{(y , v) ∈ Rk+ × Zq : Wy + Hv − b ∈ K}

), A =

[W , −Id

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

{(y , v) ∈ Rk+ × Zq : Wy + Hv − b ∈ K}

), A =

[W , −Id

Some Notation

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

A∗ denotes the conjugate linear map from Rm to EKer(A) := {x ∈ E : Ax = 0}Im(A) := {Ax : x ∈ E}

Some Notation

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

K ⊂ E is a full-dimensional, closed, convex and pointed coneand its dual cone is given by

K∗ := {y ∈ E : 〈y , x〉 ≥ 0 ∀x ∈ K} .

Some Notation

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

K∗ := the cone dual to KExt(K) := the set of extreme rays of K

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

We are interested in conv(S(A,K,B))

We are interested in conv(S(A,K,B)) conv(S(A,K,B))

conv(S(A,K,B)) = intersection of all linear valid inequalities (v.i.)〈µ, x〉 ≥ η0 for S(A,K,B):

C (A,K,B) = convex cone of all linear valid inequalities for S(A,K,B)

= {(µ; η0) : µ ∈ E , µ 6= 0,−∞ < η0 ≤ infx∈S(A,K,B)

〈µ, x〉︸︷︷︸:=µ0

Goal: Study C (A,K,B) in order to characterize the properties of the linear v.i.,and identify the necessary and/or sufficient ones defining conv(S(A,K,B))

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

= {(µ; η0) : µ ∈ E , µ 6= 0,−∞ < η0 ≤ infx∈S(A,K,B)

〈µ, x〉︸︷︷︸:=µ0

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

= {(µ; η0) : µ ∈ E , µ 6= 0,−∞ < η0 ≤ infx∈S(A,K,B)

〈µ, x〉︸︷︷︸:=µ0

Which Inequalities Should We Really Care About?

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

C (A,K,B) = convex cone of all linear valid inequalities (v.i.)

C (A,K,B)

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

⇒ Cone implied inequality, (δ; 0) for any δ ∈ K∗ \ {0}, is always valid.

C (A,K,B)

Cone impliedinequalities

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

Definition

An inequality (µ; η0) ∈ C (A,K,B) is a K-minimal valid inequality if for allρ such that ρ �K∗ µ and ρ 6= µ, we have ρ0 < η0.

Cm(A,K,B) = cone of K-minimal valid inequalities

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

Definition

S(A,K,B) = {x ∈ E : Ax ∈ B, x ∈ K}

C (A,K,B)

Cm(A,K,B)Cone impliedinequalities

K-minimal Inequalities

Definition

⇒ A K-minimal v.i. (µ; η0) cannot be written as a sum of a cone impliedinequality and another valid inequality.

⇒ Cone implied inequality (δ; 0) for any δ ∈ K∗ \ {0} is never minimal.

⇒ K-minimal v.i. exists if and only if 6 ∃ valid equations of form〈δ, x〉 = 0 with δ ∈ K∗ \ {0}.

Definition

Cone implied inequalities (δ; 0) with δ ∈ K∗ \ {0}are always valid but are never K-minimal.

yet they can still be extremal in the cone C (A,K,B).

are not particularly interesting as they are already included in thedescription (x ∈ K).

C (A,K,B)

Theorem: [Sufficiency of K -minimal Inequalities]

Whenever there is at least one K-minimal v.i., Cm(A,K,B) 6= ∅,i.e., 6 ∃δ ∈ K∗ \ {0} such that 〈δ, x〉 = 0 for all x ∈ S(A,K,B),then C (A,K,B) is generated by K-minimal v.i. and cone implied v.i.

Necessary Conditions for K-minimality

What can we say about µ for a v.i. (µ; η0) ∈ Cm(A,K,B)?

Proposition

If (µ; η0) ∈ Cm(A,K,B), then for all linear maps Z : K → K s.t.AZ ∗ ≡ A, we have µ− Zµ 6∈ K∗ \ {0}.

Proposition

More on

FK := {(Z : E → E ) : Z is linear, and Z ∗v ∈ K ∀v ∈ K}

FK is the cone of K −K positive maps.

They also appear in applications of robust optimization, quantumphysics, ...

Proposition

More on

When K = Rn+, FK = {Z ∈ Rn×n : Zij ≥ 0 ∀i , j}

When K = Ln, FK has a semidefinite programming representation(Hildebrand 2006, 2008).[

In particular when Z = abT with a, b ∈ Ln, then Z ∈ FLn .]

Proposition

More on

When K = Rn+, FK = {Z ∈ Rn×n : Zij ≥ 0 ∀i , j}

When K = Ln, FK has a semidefinite programming representation(Hildebrand 2006, 2008).[

In particular when Z = abT with a, b ∈ Ln, then Z ∈ FLn .]

Proposition

More on

But in general they are hard to describe:[Deciding whether a given linear map takes Sn

+ to itself is an NP-hard problem.

(Ben Tal & Nemirovski, 1998)

Valid Inequalities

C (A,K,B)

Cm(A,K,B)

Valid Inequalities

C (A,K,B)

Cm(A,K,B)

K-sublinear Inequalities

Definition

(µ; η0) is a K-sublinear v.i. if it satisfies

(A.1) 0 ≤ 〈µ, u〉 for all u s.t. Au = 0 and

〈α, v〉u + v ∈ K ∀v ∈ Ext(K) holds for some α ∈ Ext(K∗),(A.2) η0 ≤ 〈µ, x〉 for all x ∈ S(A,K,B).

Proposition

Any valid inequality (µ; η0) ∈ C (A,K,B) satisfies condition (A.0).

Definition

Condition (A.1) implies

(A.0) 0 ≤ 〈µ, u〉 for all u ∈ K such that Au = 0.

Proposition

Definition

(A.0) µ ∈ Im(A∗) +K∗

Proposition

Definition

(A.0) µ ∈ Im(A∗) +K∗

Proposition

Theorem

Cm(A,K,B) ⊆ Ca(A,K,B).

C (A,K,B)

Cm(A,K,B)

Ca(A,K,B)

Theorem

Cm(A,K,B) ⊆ Ca(A,K,B).

Theorem [K.-K. & Steffy]: Sufficiency of K -sublinear Inequalities

C (A,K,B) is generated by K-sublinear v.i. and cone implied v.i. withoutany restrictions on set S(A,K,B).

C (A,K,B)

Cm(A,K,B)

Ca(A,K,B)

This is all fine but the definition of K-sublinearity is much less apparent,how can we hope to verify it?

Relations to Support Functions

Consider any v.i. (µ; η0) and let Dµ = {λ ∈ Rm : µ− A∗λ ∈ K∗}.

Remark

Let σD(·) be the support function of Dµ, i.e.,

σD(h) = supλ{〈h, λ〉 : λ ∈ Dµ}.

For all v.i. (µ; η0), we have

Dµ 6= ∅,σD(0) = 0, and

σD(Ad) ≤ 〈µ, d〉 for all d ∈ K.

Necessary Conditions for K-sublinearity

Remark

For any µ ∈ Im(A∗) +K∗, ⇒ Dµ 6= ∅⇒ σD(Az) ≤ 〈µ, z〉 holds ∀z ∈ K; and

⇒ for any η0 ≤ infb∈B σD(b), we have (µ; η0) ∈ C (A,K,B).

Proposition [Necessary Condition for K-Sublinearity]

Suppose µ satisfies condition (A.0), i.e., µ ∈ Im(A∗) +K∗. Then

∀µ ∈ Im(A∗), we have σD(Az) = 〈µ, z〉 for all z ∈ K.

For any (µ; η0) ∈ C (A,K,B), if S(A,K,B) is closed, then there existsz ∈ Ext(K) s.t. σD(Az) = 〈µ, z〉.

Necessary Conditions for K-sublinearity

Remark

For any µ ∈ Im(A∗) +K∗, ⇒ Dµ 6= ∅⇒ σD(Az) ≤ 〈µ, z〉 holds ∀z ∈ K; and

⇒ for any η0 ≤ infb∈B σD(b), we have (µ; η0) ∈ C (A,K,B).

Proposition [Necessary Condition for K-Sublinearity]

Suppose µ satisfies condition (A.0), i.e., µ ∈ Im(A∗) +K∗. Then

∀µ ∈ Im(A∗), we have σD(Az) = 〈µ, z〉 for all z ∈ K.

For any (µ; η0) ∈ C (A,K,B), if S(A,K,B) is closed, then there existsz ∈ Ext(K) s.t. σD(Az) = 〈µ, z〉.

Sufficient Condition for K-sublinearity

Proposition [Sufficient Condition for K-Sublinearity]

Let (µ; η0) ∈ C (A,K,B) and suppose that ∃x i ∈ Ext(K) s.t.

σD(Ax i ) = 〈µ, x i 〉 for all i ∈ I and∑

i∈I xi ∈ int(K),

then (µ; η0) ∈ Ca(A,K,B).

Refinement of K-sublinearity for K = Rn+

Refinement of the condition (A.1) for K-sublinearity leads to

(A.0) 0 ≤ 〈µ, u〉 for all u ∈ Rn+ such that Au = 0, and

(A.1i) for all i = 1, . . . , n,

µi ≤ 〈µ, u〉 for all u ∈ Rn+ such that Au = Ai

⇒ Hence for K = Rn+, we obtain identically the class of subadditive v.i. defined

by Johnson 1981 via a much simplified analysis.

⇒ In fact Johnson 1981 also shows that one needs to verify only a finitelymany of these requirements (A.1i) for u satisfying a minimal dependencecondition.

Proposition [Necessary Condition for Rn+-Sublinearity]

Let K = Rn+. For any (µ; η0) ∈ Ca(A,K,B), we have σD(Az) = 〈µ, z〉 for all

z ∈ Ext(K), i.e., σD(Ai ) = µi for all i = 1, . . . , n where Ai is the i th column ofthe matrix A.

(A.1i) for all i = 1, . . . , n,

⇒ Complete characterization of Rn+-sublinear inequalities:

All Rn+-sublinear inequalities are generated by sublinear (subadditive and

positively homogeneous, in fact also piecewise linear and convex) functions, i.e.,support functions σDµ(·) of Dµ.[

This recovers a number of results from Johnson ’81, and Conforti et al.’13.]

⇒ This is the underlying source of a cut generating function view forlinear MIPs.

Sufficient Condition for K-minimality

Theorem [Sufficient Condition for K-Minimality]

Suppose that Cm(A,K,B) 6= ∅. Consider any (µ; η0) ∈ Ca(A,K,B) s.t.

η0 = infb∈B σD(b) and

∃x i ∈ K s.t.∑

i xi ∈ int(K), Ax i = bi with bi ∈ B satisfying

σD(bi ) = η0 and 〈µ, x i 〉 = η0,

then (µ; η0) ∈ Cm(A,K,B).

For general cones K other than Rn+, unfortunately there is a gap between

the current necessary condition and the sufficient condition forK-minimality.

Sufficient Condition for K-minimality

Theorem [Sufficient Condition for K-Minimality]

Suppose that Cm(A,K,B) 6= ∅. Consider any (µ; η0) ∈ Ca(A,K,B) s.t.

η0 = infb∈B σD(b) and

∃x i ∈ K s.t.∑

i xi ∈ int(K), Ax i = bi with bi ∈ B satisfying

σD(bi ) = η0 and 〈µ, x i 〉 = η0,

then (µ; η0) ∈ Cm(A,K,B).

For general cones K other than Rn+, unfortunately there is a gap between

the current necessary condition and the sufficient condition forK-minimality.

A Simple Example

K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e.,

S(A,K,B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥√x2

1 + x22}

(0, 0)

conv(S(A,K,B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥√

1 + x22}

A Simple Example

K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e.,

S(A,K,B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥√x2

1 + x22}

conv(S(A,K,B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥√

1 + x22}

K-minimal inequalities are:

(a) µ(+) = (1; 0; 0) with η(+)0 = −1 and µ(−) = (−1; 0; 0) with η

(−)0 = −1;

(b) µ(t) = (0; t;√t2 + 1) with η

(t)0 = 1 for all t ∈ R.

(these can be expressed as a single conic inequality x3 ≥√

1 + x22 .)

Linear inequalities in (b) cannot be generated by any cut generating function ρ(·),

i.e., ρ(Ai ) = µ(t)i is not possible for any function ρ(·).

One cannot hope to develop a strong conic dual for problems of form

minx{〈c , x〉 : Ax = b, x ∈ K, xi is integer for all i ∈ I}.

[Sharp contrast to the results of Dey, Moran & Vielma ’12. ]

A Simple Example

K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e.,

S(A,K,B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥√x2

1 + x22}

conv(S(A,K,B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥√

1 + x22}

(−)0 = −1;

(b) µ(t) = (0; t;√t2 + 1) with η

(t)0 = 1 for all t ∈ R.

1 + x22 .)

A Simple Example

K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e.,

S(A,K,B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥√x2

1 + x22}

conv(S(A,K,B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥√

1 + x22}

(−)0 = −1;

(b) µ(t) = (0; t;√t2 + 1) with η

(t)0 = 1 for all t ∈ R.

1 + x22 .)

A Simple Example

K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e.,

S(A,K,B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥√x2

1 + x22}

conv(S(A,K,B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥√

1 + x22}

(−)0 = −1;

(b) µ(t) = (0; t;√t2 + 1) with η

(t)0 = 1 for all t ∈ R.

1 + x22 .)

Can we derive conic valid inequalities for S(A,K,B) using this framework?

Disjunctive Cuts for Lorentz Cone, Ln

Start with a simple set for x , i.e., x ∈ K = Ln

Consider a two-term disjunction of formeither πT1 x ≥ π1,0 or πT2 x ≥ π2,0 must hold.

Let Si := {x : πTi x ≥ πi ,0, x ∈ K}.

Without loss of generality assume that π1,0, π2,0 ∈ {0,±1} and S1 6= ∅ and S2 6= ∅.

Let Si := {x : πTi x ≥ πi ,0, x ∈ K}.By setting

[πT1πT2

], and B =

{[π1,0 + R+

]⋃[R

π2,0 + R+

]}we arrive back at

S(A,K,B) = {x ∈ Rn : Ax ∈ B, x ∈ K}

and S(A,K,B) = S1 ∪ S2.

The cases of S1 ⊆ S2 or S2 ⊆ S1 are not interesting, so we assume

Assumption

The disjunction πT1 x ≥ π1,0 and πT2 x ≥ π2,0 satisfy

{β ∈ Rn+ : βπ1,0 ≥ π2,0, π2 − βπ1 ∈ Ln} = ∅, and

{β ∈ Rn+ : βπ2,0 ≥ π1,0, π1 − βπ2 ∈ Ln} = ∅.

Convex nonlinear valid inequalities for Ln via a disjunctive argument[Sketch of derivation]:

Given a disjunction πT1 x ≥ π1,0 and πT

2 x ≥ π2,0 with π1,0, π2,0 ∈ {0,±1}

Characterize the structure of linear Ln-minimal valid inequalities

Based on their characterization, e.g., (β1, β2) values, group all of the linearLn-minimal valid linear inequalities via an optimization problem over µ

Turns out to be a nonconvex optimization problem, but it has a tightrelaxation

Process the relaxation of this problem by taking its dual, etc., to arrive atthe following main result

2 x ≥ π2,0 with π1,0, π2,0 ∈ {0,±1}Characterize the structure of linear Ln-minimal valid inequalities

Proposition [K.-K., Yıldız]

For all Ln-minimal valid linear inequalities µ>x ≥ µ0 for conv(S1 ∪ S2) thereexists α1, α2 ∈ bd(Ln), and β1, β2 ∈ (R+ \ {0}) s.t.

µ = α1 + β1π1,

µ = α2 + β2π2,

min{π1,0β1, π2,0β2} = µ0 = min{π1,0, π2,0},

and at least one of β1 and β2 is equal to 1.

2 x ≥ π2,0 with π1,0, π2,0 ∈ {0,±1}

Disjunction: either πT1 x ≥ π1,0 or πT

2 x ≥ π2,0

Theorem [K.-K., Yıldız]

Let σ = min{π1,0, π2,0}. For any β > 0 such that βπ1 − π2 /∈ ±int(Ln), thefollowing convex inequality is valid for conv(S1 ∪ S2):

2σ − (βπ1 + π2)>x ≤√

((βπ1 − π2)>x)2

+ N(β) ∗ (x2n − ‖x‖2

where N(β) := ‖βπ1 − π2‖22 − (βπ1,n − π2,n)2, and

exactly captures all linear v.i. corresponding to β1 = β and β2 = 1.

2σ − (βπ1 + π2)>x ≤√

((βπ1 − π2)>x)2

+ N(β) ∗ (x2n − ‖x‖2

In certain cases such as

conv(S1 ∪ S2) is closed,

Splits, i.e., π1 = −απ2 for some α > 0, and π1,0 = π2,0 = σ with σ = 1

it is sufficient (for conv(S1 ∪ S2)) to consider only one inequality with β = 1.

For splits with rhs σ = 1, it is exactly the following conic quadratic inequality∥∥∥∥x − 2(π>1 x − σ)

N(1)(π1 − π2)

∥∥∥∥2

≤(xn +

2(π>1 x − σ)

N(1)(π1,n − π2,n)

2σ − (βπ1 + π2)>x ≤√

((βπ1 − π2)>x)2

+ N(β) ∗ (x2n − ‖x‖2

In certain cases such as

conv(S1 ∪ S2) is closed,

Splits, i.e., π1 = −απ2 for some α > 0, and π1,0 = π2,0 = σ with σ = 1

it is sufficient (for conv(S1 ∪ S2)) to consider only one inequality with β = 1.

For splits with rhs σ = 1, it is exactly the following conic quadratic inequality∥∥∥∥x − 2(π>1 x − σ)

N(1)(π1 − π2)

∥∥∥∥2

≤(xn +

2(π>1 x − σ)

N(1)(π1,n − π2,n)

Disjunction: x3 ≥ 1 or 2x1 + 2x3 ≥ 1

conv(S1 ∪ S2) =

{x ∈ L3 : 2− (2x1 + 3x2) ≤

√(−2x1 − x3)2 + 3 (x2

3 − x21 − x2

}F. Kılınc-Karzan (CMU) Structure of Valid Inequalities for MICPs 32 / 34

Disjunction: x3 ≥ 1 or 2x1 + 2x3 ≥ 1

conv(S1 ∪ S2) =

{x ∈ L3 : 2− (2x1 + 3x2) ≤

√(−2x1 − x3)2 + 3 (x2

3 − x21 − x2

Disjunction: −x3 ≥ −1 or −x2 ≥ 0

conv(S1 ∪ S2) ={x ∈ L3 : x2 ≤ 1, 1 + |x1| − x3 ≤

√1−max{0, x2}2

Final remarks

Introduce a unifying framework for defining K-minimal andK-sublinear inequalities for conic MIPs

Understand when K-minimal inequalities exist and are sufficient

Characterize structure of K-minimal and K-sublinear inequalities

Necessary, and also sufficient conditionsRelation with support functions of certain structured sets

Captures previous results from the MIP literature, i.e., K = Rn+

(i.e., Johnson ’81 and Conforti et al.’13)

For K = Ln, by studying structure of linear K-minimal inequalitiesfrom this framework, we derive explicit expressions for conic cuts

Covers most of the recent results on conic MIR, split, and two-termdisjunctive inequalities (i.e., Belotti et al.’11, Andersen & Jensen ’13,

Modaresi et al.’13)Much more intuitive and elegant derivations covering new cases

Final remarks

Thank you!

Structure of Valid Inequalities for Mixed Integer Conic Programs · 2014. 1. 6. · Mixed Integer...

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