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Proceedings of GO 2005, pp. 165 – 170.Parametrical approach for studying and solving bilinearprogramming problemDmitrii LozovanuInstitute of Mathematics and Computer Science of Moldovan Academy of Sciences, Academy str., 5, Chisinau, MD–2028lozovanu@math.mdAbstractThe linear parametrical programming approach for studying and solving a bilinear programmingproblem (BPP) is proposed. On the basis of the duality theory BPP is transformed into a problemof determining compatibility of a system of linear inequalities with a right part that depends onparameters, admissible values of which are determined by an another system of linear inequalities.Some properties of this auxiliary problem are obtained and algorithms for solving BPP are proposed.Keywords:Bilinear Programming, Linear Parametrical Systems, Duality Principle for Parametrical Systems1. Introduction and preliminary resultsWe consider the following bilinear programming problem (BPP) [1,2]:to minimize the object functionz = xCy + d 1 x + d 2 y (1)on subjectA 1 x ≤ b 1 , x ≥ 0; (2)A 2 y ≤ b 2 , y ≥ 0, (3)where C, A 1 , A 2 are matrices of size n×m, m×n, k×m, respectively, and d 1 , x ∈ R n ; d 2 , y, b 1 ∈R m ; b 2 ∈ R k . This problem can be solved by varying the parameter h ∈ [−2 L , 2 L ] in theproblem of determining compatibility of the system⎧⎨⎩A 1 x ≤ b 1 , x ≥ 0;xCy + d 1 x + d 2 y ≤ h;A 2 y ≤ b 2 , y ≥ 0,where L is the size of the problem (1)-(3).Furthermore we regard the compatibility problem with respect to x of the system(4){A 1 x ≤ b 1 , x ≥ 0;xCy + d 1 x ≤ h − d 2 y.(5)for every y satisfying (3).The following Theorems are proved in [3] on the basis of the linear duality theory appliedto (5).