OPERATIONS RESEARCH LECTURE NOTES

“INTRODUCTION TO INTERIOR POINT METHODS”

Assoc. Prof. Dr. Y. İlker Topcu

Acknowledgements: I would like to acknowledge Prof. W.L. Winston's "Operations Research: Applications and Algorithms“ (slides submitted by Brooks/Cole, a division of Thomson Learning, Inc.) and Prof. J.E. Beasley's lecture notes which greatly influence these notes... I retain responsibility for all errors and would love to hear from readers... www.isl.itu.edu.tr/ya

Assoc. Prof. Dr. Y. İlker Topcu (www.ilkertopcu.net)

INTRODUCTION TO INTERIOR POINT METHODS

Interior Point Methods (IPM) still follow the improving search paradigm for LP, but they

employ moves quite different from those in simplex method.

Much more effort turns out to be required per move (iteration) with IPM but the number

of moves decreases dramatically.

The simplex algorithm is an exponential time algorithm for solving LPs.

If an LP of size n is solved by the simplex, then there exists a positive number a

such that for any n, the simplex algorithm will find the optimal solution in a time of

at most a2n.

IPM, on the other hand, is a polynomial time algorithm.

This implies that if an LP of size n is solved by an IPM, then there exist positive

numbers b and c such that for any n, LP can be solved in a time of at most bnc

Instead of staying on the boundary of the feasible region and passing from extreme point

to extreme point, IPM proceed directly across the interior.

IPM begin at and move through a sequence of interior feasible solutions, converging to

the boundary of the feasible region only at an optimal solution.

Popular methods

• Karmarkar’s Projective Transformation

• Affine Scaling

• Log Barrier

IPM are applied to an LP in the following standard form:

max c · x s.t. Ax = b

x ≥ 0

Assoc. Prof. Dr. Y. İlker Topcu (www.ilkertopcu.net) 1

Interior in LP Standard Form A feasible solution for an LP in standard form is an interior point if every component

(variable) of the solution that can be positive in any feasible solution is strictly positive in

the given point.

Projecting to Deal with Equality Constraints A move direction ∆x is feasible for equality constraints Ax=b if it satisfies A∆x=0.

The projection of a move vector d on a given system of equalities is a direction

preserving those constraints and minimizing the total squared difference between its

components and those of d.

The projection of direction d onto conditions A∆x=0 preserving linear inequalities Ax=b

can be computed as ∆x=Pd where projection matrix P = I – AT (AAT)–1 A

Improvement with Projected Directions The projection ∆x= Pc of (nonzero) objective function vector c onto equality constraints

Ax=b is an improving direction at every x.

±

Affine Scaling Approach (x1’, x2’, x3’): initial solution

Diagonal matrix D = ⎢⎥⎥⎥

⎦

⎤

⎢

⎢

⎣

⎡

'000'000'

3

2

1

xx

x

xDx ~=such that

Rescaled variable: [Centering scheme (1, 1, 1)] Dx 1~ −= xFor new coordinates

ADA =~

Dcc =~

A)AA(AIP ~~~~ 1TT −−=Projection matrix:

cPc ~P ±=Projected gradient:

Assoc. Prof. Dr. Y. İlker Topcu (www.ilkertopcu.net) 2

Pnew~~ cxx

vα

+=Update

where v is the absolute value of the negative component of cp having largest

absolute value and a is arbitrarily chosen as 0.8

α measures the fraction used of the distance that could be moved before the

feasible region is left. An α value close to upper bound of 1 is good for giving a

relatively large step toward optimality on the current iteration. However, the

problem with a value too close to 1 is that the next trial solution then is jammed

against a constraint boundary, thereby making it difficult to take large improving

steps during subsequent iterations.

newnew~ xDx =In original coordinates:

When the solution value is no longer changing very much, algorithm stops

Example 1. Frannie’s Firewood (Rardin 6.1., p. 274)

max 90 x1 + 150 x2

s.t. 0.5 x1 + x2 ≤ 3

x1, x2 ≥ 0

Please refer to www.isl.itu.edu.tr/ya/interior.htm for the answer

Assoc. Prof. Dr. Y. İlker Topcu (www.ilkertopcu.net) 3