Geodesic
A Linear Parametrical Programming Approach for Studying and Solving Bilinear Programming Problem
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2006), pp. 73-86 
Dmitrii Lozovanu; Maria Fonoberova. A Linear Parametrical Programming Approach for Studying and Solving Bilinear Programming Problem. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2006), pp. 73-86. http://geodesic.mathdoc.fr/item/BASM_2006_3_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

An approach for studying and solving a bilinear programming problem, based on linear parametrical programming, is proposed. Using duality principle for the considered problem we show that it can be transformed into a problem of determining the compatibility of a system of linear inequalities with a right-hand member that depends on parameters, admissible values of which are defined by another system of linear inequalities. Some properties of this auxiliary problem are obtained and a conical algorithm for its solving is proposed. We show that this algorithm can be used for finding the exact solution of bilinear programming problem as well as its approximate solution.

[1] Altman M., “Bilinear Programming”, Bul. Acad. Polan. Sci., Ser. Sci. Math. Astron. et Phis., 1968, no. 16(9), 741–746 | MR | Zbl

[2] Chernicov S. N., Linear Inequalities, Nauka, Moscow, 1968 | MR

[3] Horst R., Tuy H., Global Optimization. Deterministic Approaches, Springer Verlag, Berlin, 1993 | MR

[4] Khachian L. G., “Polynomial time algorithm in linear programming”, USSR, Computational Mathematics and Mathematical Physics, 20 (1980), 51–58 | DOI

[5] Khachian L. G., “On exact solution of the system of linear inequalities and linear programming problem”, USSR, Computational Mathematics and Mathematical Physics, 22 (1982), 999–1002 | MR

[6] Lozovanu D., “Duality Principle for Systems of Linear Inequalities with a Right-Hand Member that Depends on Parameters”, Izv. AN SSSR, Ser. Techn. Cyb., 6 (1987), 3–13 (in Russian)

[7] Lozovanu D., Extremal-Combinatorial Problems and Algorithms for their Solving, Stiintsa, Chisinau, 1991 (in Russian)

[8] Lozovanu D., “Parametrical approach for studying and solving bilinear programming problem”, Proceedings of the International Workshop on Global Optimization (San Jose, Almeria, Spain, September 18–22th, 2005), 165–170

[9] Pardalos P., Rosen J., “Methods for Global Concave Minimization. A bibliographical Survey”, SIAM Review, 1986, no. 28(3), 367–379 | DOI | MR | Zbl

[10] Tuy H., “Normal Conical Algorithm for Concave Minimization over Polytopes”, Math. Progr., 51 (1991), 229–245 | DOI | MR | Zbl