View - Universidad de Almería

Proceedings of GO 2005, pp. 195 – 198.On solving of bilinear programming problems ∗Andrey V. OrlovInstitute of System Dynamics and Control Theory, Laboratory of Global Optimization Methods, Lermontov st. 134, Irkutsk,Russia, anor@icc.ruAbstractKeywords:In this paper we suggest a new approach for solving disjoint bilinear programming problems. Thisapproach is based on the Global Search Strategy for d.c. maximization problems, developed byA.S.Strekalovsky. Two local search methods and outline of global search algorithm for such problemsare presented.bilinear programming, d.c. functions, special methods of local search, bimatrix games, global search1. IntroductionA number problems in engineernig design, decision theory, operation research and economycan be described by the bilinear programming problems. For example, finding the Nash equilibriumin bimatrix games [3, 4] and bilinear separation problem in IR n [5] can be formulatedas specific bilinear problems.In spite of external simplicity such problems are nonconvex. As is well known in nonconvexproblems there exist a lot of local extremums or even stationary (critical) points, thatare different from global solutions. And classical methods of convex optimization [1] are notapplicable in such problems.There are two types of bilinear programs: with joint constraints and with disjoint constraints.The former is more hard problem than the latter. But even for problems with disjointconstraints development of fast algorithm is complicated problem. Several methodshave been proposed in literature to solve jisjoint bilinear problems [6]). The aim of this paperis to study the usefullness of Global Optimality Conditions approach, developed by A.S.Strekalovsky [2], for disjoint bilinear programs.2. Problem statement and d.c. decomposition of goal functionLet us consider the bilinear functionF (x, y) = 〈c, x〉 + 〈x, Qy〉 + 〈d, y〉, (1)where c, x ∈ IR m ; d, y ∈ IR n ; Q is an (m × n) matrix.The disjoint bilinear programming problem can be written as follows:F (x, y) ↑ max ,⎫(x,y) ⎪⎬s.t. x ∈ X △ = {x ∈ IR m | Ax ≤ a, x ≥ 0},y ∈ Y △ = {y ∈ IR n | By ≤ b, y ≥ 0},⎪⎭(BLP )∗ This work was supported by RFBR Grant No. 05-01-00110