Abstract
This paper addresses the optimistic statement of one class of bilevel optimization problems (BOPs) with a nonconvex lower level. Namely, we study BOPs with a convex quadratic objective function at the upper level and with a bimatrix game at the lower level. It is known that the problem of finding a Nash equilibrium point in a bimatrix game is equivalent to the special nonconvex optimization problem with a bilinear structure. Nevertheless, we can replace such a lower level with its optimality conditions and transform the original bilevel problem into a single-level nonconvex optimization problem. Then we apply the original Global Search Theory (GST) for general D.C. optimization problems and the Exact Penalization Theory to the resulting problem. After that, a special method of local search, which takes into account the structure of the problem under consideration, is developed.
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Acknowledgement
The research was funded by the Ministry of Education and Science of the Russian Federation within the framework of the project “Theoretical foundations, methods and high-performance algorithms for continuous and discrete optimization to support interdisciplinary research” (No. of state registration: 121041300065-9).
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Orlov, A.V. (2021). On Solving Bilevel Optimization Problems with a Nonconvex Lower Level: The Case of a Bimatrix Game. In: Pardalos, P., Khachay, M., Kazakov, A. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2021. Lecture Notes in Computer Science(), vol 12755. Springer, Cham. https://doi.org/10.1007/978-3-030-77876-7_16
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