Combined relaxation methods for finding equilibrium points and solving related problems.

*(English. Russian original)*Zbl 0835.90123
Russ. Math. 37, No. 2, 44-51 (1993); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 2, 46-53 (1993).

Summary: The problem to find an equilibrium point is one of the main problems in game theory and its applications, especially in economics. It also includes well-known problem of finding a saddle point of Lagrange function. The equilibrium problem is closely connected with the problem of variational inequalities solving, which, in its turn, has a lot of other applications. Hence, methods for solving both the problems are intensively investigated. It should be noted that both methods, being constructed in analogy to well-known optimization methods, do not guarantee convergence, if the strict convex-concavity of the corresponding function or the strict antimonotonicity of the respective mapping are not provided.

On the other hand, even sophisticated methods, similar to the Brown- Volkonskii one, converge rather slowly. There are certain ways to overcome the indicated drawbacks. One of these is to regularize the original problem so that the needed properties may hold for it. Furthermore, methods with averaging of directions and the extragradient method ensure convergence without any modification of the original problem. In the present article we propose methods based on previous research of the author. It is shown that the assumption of smoothness allows us to construct simple realizations which do not use a priori information on the problem.

On the other hand, even sophisticated methods, similar to the Brown- Volkonskii one, converge rather slowly. There are certain ways to overcome the indicated drawbacks. One of these is to regularize the original problem so that the needed properties may hold for it. Furthermore, methods with averaging of directions and the extragradient method ensure convergence without any modification of the original problem. In the present article we propose methods based on previous research of the author. It is shown that the assumption of smoothness allows us to construct simple realizations which do not use a priori information on the problem.