A Primal-Dual Algorithm for Multiobjective Linear Programming

27 Congreso Nacional de Estad´ıstica e Investigaci´ on Operativa Lleida, 8–11 de abril de 2003

A Primal-Dual Algorithm for Multiobjective Linear Programming M. Ehrgott1 , J. Puerto2 , Antonio M. Rodr´ıguez-Ch´ıa3 , 1

Department of Engineering Science University of Auckland E-mail: [email protected]

2

Departamento de Estad´ıstica e Investigaci´on Operativa Universidad de Sevilla, 41013 Sevilla, Espa˜ na

E-mail: [email protected]

2

Departamento de Estad´ıstica e Investigaci´on Operativa

Universidad de C´adiz, 11510 Puerto Real (C´adiz) Espa˜ na E-mail: [email protected]

RESUMEN En este trabajo se desarrolla un algoritmo primal-dual para un problema de programaci´on lineal multiobjetivo. Este algoritmo est´a basado en el teorema de escalarizaci´on de soluciones eficientes para problemas de programaci´on lineal multiobjetivo y el cl´asico algoritmo primal-dual. Se analiza la complejidad de la metodolog´ıa propuesta, finalizando con la ilustraci´on de dicho algoritmo a trav´es de un ejemplo. Palabras y frases clave: Multiobjective Linear Programming, Primal-dual Algorithm. Clasificaci´ on AMS: 90C05, 90C29.

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Introduction

Primal-dual algorithms have proven to be very efficient for several classes of linear programming problems, in particular those related to network optimization problems. These include the Hungarian method for assignment problems, the augmenting path method for minimum cost flow problems, Dijsktra’s algorithm for shortest path problems, etc. (see e.g. Nemhauser and Wolsey (1988)). In this paper we introduce a primal-dual algorithm for multiobjective linear programs in order to investigate whether these features can be extended to multiobjective problems. Multiobjective linear programming has been a topic of research for about three decades. Many algorithms, often extensions of the simplex method to the multiobjective case, are known, e.g. (Evans and Steuer (1973), Isermann (1977), Steuer (1985), Yu and Zeleny (1975), Yu (1976), Zeleny (1974)). More recently, methods that focus on determination of the efficient set in outcome space, rather than the Pareto optimal set in decision space have been of interest, Benson (1997). Also, duality theory for multiobjective linear programming has been studied, e.g. Corley (1984), Isermann (1978). It is therefore surprising that no primal-dual method has been published. One of the reasons for this might be that multiobjective duality theory cannot easily be used to develop an algorithm analogous to the single objective case, due to the different structure of the complementary slackness condition. Therefore we follow a different approach and use scalarization of multiobjective linear programmes and single objective duality theory. Despite the interest in MOLP over the last three decades, a primal-dual algorithm algorithm had not been proposed before. Because we base the algorithm on duality of scalarized linear programmes P (λ), we can compute in parallel both Pareto optimal solutions and the subdivision of the parameter space. We also avoid degeneracy in the primal representation of vertices. The algorithm allows to use parallel computation to generate Pareto optimal

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solutions for each parameter λ ∈ Λ by exploiting the branching structure obtained through subdivision of the parameter space. It also allows to give another interpretation of Pareto optimal solutions as the sensitivity analysis of several right hand side vectors in scalar linear programming problems. The major disadvantage of the algorithm is that we may have to explore exponentially many nodes. But notice that this cannot be avoided by any algorithm because of the possibly exponentially many extreme points of the efficient set.

Acknowledgements The paper was written during a visit of the first author to the University of Seville financed by a grant of the Andalusian Consejeria de Educaci´on. The research of the second and third authors is also partially financed by Spanish research grants BFM2001-2379 and BFM2001-4028.

Referencias H.P. Benson (1988): 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(1):1–24. H.W. Corley (1984): Duality theory for the matrix linear programming problem. Journal of Mathematical Analysis and Applications, 104:47–52. G.L. Nemhauser and L.A. Wolsey (1988): Integer and Combinatorial Optimization. Wiley, New York. J.P. Evans and R.E. Steuer (1973): A revised simplex method for linear multiple objective programs. Mathematical Programming, 5:375–377, 1973. H. Isermann (1977): The enumeration of the set of all efficient solutions for a linear multiple objective program. Operational Research Quarterly, 28(3):711–725, 1977.

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R.E. Steuer (1985): Multiple Criteria Optimization: Theory, Computation and Application. John Wiley & Sons, New York, NY. P.L. Yu and M. Zeleny (1975): The set of all nondominated solutions in linear cases and a multicriteria simplex method. Journal of Mathematical Analysis and Applications, 49:430–468. P.L. Yu and M. Zeleny (1976): Linear multiparametric programming by multicriteria simplex method. Management Science, 23(2):159–170. M. Zeleny (1974): Linear Multiobjective Programming. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, 25.

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