Speciality:
05.13.16 (Computer techniques, mathematical modelling, and mathematical methods with an application to scientific researches)
Birth date:
10.09.1939
E-mail: ,
Website: http://www.ccas.ru/antipin/ant-r.htm Keywords: nonlinear programming; optimization methods: variational inequalities; equilibrium programming; fixed points; convetrgence; stability.
Subject:
Two new inequalities are established, first of which describes a class of strongly convex differentiable functions, second inequality links three any points of set for any convex function, alwyas supposing that gradients of these functions subjected to the Lipschitz conditions. For minimization of functions over convex sets is formulated differential (continuous) the gradient projection method of first and second order with projection operator of a point onto a permissible set. In convex case the convergence of trajectories to optimal solution is proved, estimates of convergence rate for continuous methods are given. A equilibrium programming problem is formulated, where a solution of it is a fixed point of extreme mapping. In particular, a equilibrium problem includes a n-person game with Nash equilibrium. It is shown that the equilibrium problem can always be splitted on a sum of two problems one of which is saddle problem and other is a optimiztion one. New inequality is offered, with the help of which it is possible to describe the positive semi-definite class of equilibrium problems. The theory of methods to compute fixed points of this class problems is developed. The theory offered includes extragradient and extraproximal approaches, Newton-type methods and regularization and penalty function methods (the latter are developed in the co-authorship with F. P. Vasil'ev). It is shown that the offered theory is fundamentals to develop methods of the solution of n-person non-zero-sum games. The convergence to Nash equilibrium for two-person non-zero-sum game for extragradient and extraproximal methods are proved.
Main publications:
Antipin A. Gradient approach of computing fixed points of equilibrium problems // Journal of Global Optimization. 2001, 1–25.
Antipin A. Gradient-Type Method for Equilibrium Programming Problems with Coupled Constraints // Yugoslav Journal of Operations research. 2000. V. 10, no. 2, 1–15.
Antipin A. Differential equations for equilibrium problems with coupled constraints // Nonlinear Analysis, 2001, V. 47, 1833–1844.
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