A Dual Variant of Benson s Outer Approximation Algorithm for Multiple Objective Linear Programs Matthias Ehrgott 1, Andreas Löhne 2 und Lizhen Shao 3 1,3 University of Auckland, New Zealand 2 Martin-Luther-Universität Halle-Wittenberg, Germany
Outline 1. The Idea of Benson s Algorithm 2. Geometric Duality 3. The Dual Algorithm 4. Degeneracy 5. Numerical Results
1. The Idea of Benson s Algorithm Properties of MOLP in typical applications: many variables (> 1000) many inequalities or equalities (> 1000) just a few criteria (< 10) Consequences Simlplex like methods are not appropriate Construction in the image space is preferable
1. The Idea of Benson s Algorithm Properties of MOLP in typical applications: many variables (> 1000) many inequalities or equalities (> 1000) just a few criteria (< 10) Consequences Simlplex like methods are not appropriate Construction in the image space is preferable
1. The Idea of Benson s Algorithm The primal MOLP: A R m n, b R m, P R q n (P) P x v-min, s.t. Ax b Notation: feasible set: X := {x R m Ax b} (assumed to be nonempty) image set: P [X] := x X P x (assumed to be R q + -bounded below) extended image set: P := P [X] + R q +
1. The Idea of Benson s Algorithm Bensons Algorithm (1998): An inequality representation and a vertex representation of the extended image set P is constructed Proposition 1 Let y wminp. Then there exists a solution of b T u y T c max s.t. (u, c) 0, A T u = P T c, (1,..., 1) T c = 1 and for each such solution (ū, c) U, H (ū, c) := () D(ū, c) = { y R q y T c = b T ū } is a supporting hyperplane to P with y H(ū, c).
1. The Idea of Benson s Algorithm P [X]
1. The Idea of Benson s Algorithm P P [X]
1. The Idea of Benson s Algorithm P p P [X]
1. The Idea of Benson s Algorithm S 0 p P [X]
1. The Idea of Benson s Algorithm S 0 p P [X] s 1
1. The Idea of Benson s Algorithm S 0 p P [X] s 1
1. The Idea of Benson s Algorithm S 0 p P [X] y 1 s 1
1. The Idea of Benson s Algorithm S 0 p P [X] y 1 s 1 H(u 1, c 1 )
1. The Idea of Benson s Algorithm S 1 p P [X]
1. The Idea of Benson s Algorithm S 1 p P [X] s 2
1. The Idea of Benson s Algorithm S 1 p P [X] s 2
1. The Idea of Benson s Algorithm S 1 p y 2 P [X] s 2
1. The Idea of Benson s Algorithm S 1 p y 2 P [X] s 2 H(u 2, c 2 )
1. The Idea of Benson s Algorithm S 2 p P [X]
1. The Idea of Benson s Algorithm S 2 p P [X] s 3
1. The Idea of Benson s Algorithm S 2 p P [X] s 3
1. The Idea of Benson s Algorithm S 2 p P [X] y 3 s 3
1. The Idea of Benson s Algorithm S 2 p P [X] y 3 s 3 H(u 3, c 3 )
1. The Idea of Benson s Algorithm S 3 p P [X]
2. Geometric Duality
2. Geometric Duality primal image set dual image set P [X] D[U] Geometric Duality Theorem. There is an inclusion reversing oneto-one map between the set of all proper K-maximal faces of D and the set of all proper weakly minimal faces of P.
2. Geometric Duality primal extended image set dual extended image set P = P [X] + R q + P [X] D[U] D = D[U] K Geometric Duality Theorem. There is an inclusion reversing oneto-one map between the set of all proper K-maximal faces of D and the set of all proper weakly minimal faces of P.
2. Geometric Duality primal extended image set dual extended image set P = P [X] + R q + P [X] D[U] D = D[U] K Geometric Duality Theorem. There is an inclusion reversing oneto-one map between the set of all proper K-maximal faces of D and the set of all proper weakly minimal faces of P.
3. The Dual Algorithm
3. The Dual Algorithm The dual variant of Bensons Algorithm: An inequality representation and a vertex representation of the extended image set D is constructed By geometric duality we immediately obtain an inequality representation and a vertex representation of P, too. Proposition 2. Let v Max K D, then for each solution x of c(v) T P x min s.t. Ax b H ( x) :=
3. The Dual Algorithm D[U]
3. The Dual Algorithm D[U] D
3. The Dual Algorithm D[U] d D
3. The Dual Algorithm D[U] d S 0
3. The Dual Algorithm D[U] d S 0
3. The Dual Algorithm v 0 D[U] d S 0
3. The Dual Algorithm v 0 D[U] H (x 0 ) d S 0
3. The Dual Algorithm D[U] d S 1
3. The Dual Algorithm s 1 D[U] d S 1
3. The Dual Algorithm s 1 D[U] d S 1
3. The Dual Algorithm s 1 v 1 D[U] d S 1
3. The Dual Algorithm s 1 v 1 H (x 1 ) D[U] d S 1
3. The Dual Algorithm D[U] d S 2
3. The Dual Algorithm s 2 D[U] d S 2
3. The Dual Algorithm s 2 D[U] d S 2
3. The Dual Algorithm s 2 v 2 D[U] d S 2
3. The Dual Algorithm s 2 v 2 D[U] H (x 2 ) d S 2
3. The Dual Algorithm D[U] d S 3
3. The Dual Algorithm s 3 D[U] d S 3
3. The Dual Algorithm s 3 D[U] d S 3
3. The Dual Algorithm s 3 D[U] v 3 d S 3
3. The Dual Algorithm s 3 D[U] v 3 H (x 3 ) d S 3
3. The Dual Algorithm D[U] d S 4
4. Degeneracy
4. Degeneracy point representation of P inequality representation of D
4. Degeneracy point representation of P inequality representation of D
4. Degeneracy Result of the primal algorithm nondegenerate point representation of P nondegenerate inequality representation of D possibly degenerate inequality representation of P possibly degenerate point representation of D
4. Degeneracy Result of the dual algorithm nondegenerate point representation of D nondegenerate inequality representation of P possibly degenerate inequality representation of D possibly degenerate point representation of P
4. Degeneracy D[U] d S 1
4. Degeneracy D[U] d S 1
4. Degeneracy D[U] d S 1
5. Numerical Results
5. Numerical results Example 1 P = 1 0 0 0 1 0 0 0 1, A = 1 1 1 1 3 1 3 4 0 1 0 0 0 1 0 0 0 1, b = 5 9 16 0 0 0. y 3 3 2 1 0 1 y 2 y 1 5 0 5 6 4 2 v 3 =b T u 0 2 4 0 0.2 0.4 0 2 0.6 v 1 0.8 1 0 0.2 0.4 v 2 0.6 0.8 1
5. Numerical results Example 2 MOLP relaxation of an assignment problem with three objectives 3 6 4 5 2 3 5 4 3 5 4 2 4 5 3 6, 2 3 5 4 5 3 4 3 5 2 6 4 4 5 2 5, 4 2 4 2 4 2 4 6 4 2 6 3 2 4 5 3. 25 15 20 v 3 =b T u 10 y 3 15 10 10 15 20 25 y 1 40 20 y 2 0 5 0 0.5 v 1 1 0 v 2 0.5 1
5. Numerical results Example 3 Radio therapie treatment planning 597 variables, 1664 inequalities 6 8 10 12 y 2 14 y 1 20 15 80 60 y 3 10 60 70 80 v 3 40 20 0 0 90 0.5 v 1 1 0 0.2 0.4 0.6 0.8 1 v 2
5. Numerical results Computation time Matlab 7.1(R14) using CPLEX 10.0 as LP solver dual processor CPU with 1.8GHZ and 1GB RAM On-line Vertex Enumeration by Adjacency Lists Method, (Chen and Hansen, 1991) size of matrix A ext. primal image time (seconds) m n vertices facets primal dual 1 16 16 4 9 0.24 0.20 2 6 3 7 11 0.30 0.20 3 1664 597 55 85 46.2 36.1
Choice of Literature Benson, H. P.: An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem, Journal of Global Optimization 13, (1998), 1 24 Chen, P. C. and Hansen, P.: On-line and off-line vertex enumeration by adjacency lists, Operations Research Letters, 10, 403-409, 1991 Heyde, F.; Löhne, A.: Geometric duality in multi-objective linear programming, submitted to SIAM Optimization, 2006 Shao, L.; Ehrgott, M.: Approximately Solving Multiobjective Linear Programmes in Objective Space and an Application in Radiotherapy Treatment Planning, submitted to Mathematical Methods of Operations Research, 2006