Title: An algorithm for solving a min-max problem by adaptive method

Authors: Aghiles Azizen; Kahina Louadj; Mohamed Aidene

Addresses: Department of Mathematics, Faculty of Science, Mouloud Mammeri University, Tizi Ouzou, 15000, Algeria ' Department of Mathematics, Faculty of Science and Applied Science, Akli Mohand Oulhadj University, Bouira, 10000, Algeria ' Department of Mathematics, Faculty of Science, Mouloud Mammeri University, Tizi Ouzou, 15000, Algeria

Abstract: Min-max problems occupies an important place in linear programming (LP), as it addresses a large number of optimisation problems, in various fields of science. In this study, an algorithm using adaptive method is proposed for solving the min-max problem in linear programming. It consists on finding the maximum of the minimum of a function (where the essential constraints are in equality and the direct constraints are bounded) in a minimum execution time. A solving algorithm is built using the principle of the adaptive method and it is based on the concept of the support matrix of the problem. Necessary and sufficient conditions for the optimality of a support feasible solution are established and sub-optimality criterion is derived. This algorithm allows to solve directly the considered problem, without modifying it and avoids the drawbacks of the increase in the number of the variables and the constraints, thus, improve the convergence speed of the method. Its performance is tested on a numerical example.

Keywords: min-max problem; linear programming; adaptive method; sub-optimality estimate; change of support; optimisation; feasible solution; optimality criterion.

DOI: 10.1504/IJMOR.2023.133718

International Journal of Mathematics in Operational Research, 2023 Vol.26 No.1, pp.96 - 110

Received: 10 Apr 2022
Accepted: 22 Jun 2022
Published online: 02 Oct 2023 *

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