A. S. Antipin

Equilibrium programming using proximal-like algorithms

We compute constrained equilibria satisfying an optimality condition. Important examples include convex programming, saddle problems, noncooperative games, and variational inequalities. Under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization. In the main algorithm each stage of computation requires two proximal steps, possibly using Bregman functions. One step serves to predict the next point; the other helps to correct the new prediction. To enhance practical applicability we tolerate numerical errors.

Gradient Approach of Computing Fixed Points of Equilibrium Problems

Potential equilibrium problems are considered. The notions of bilinear differential and bi-convexity are introduced. The concept of generalized potentiality is offered. The convergence of gradient prediction-type methods for solving of generalized potential equilibrium problems is justified. Estimates of convergence rate are derived.

Solving variational inequalities with coupling constraints with the use of differential equations

where F (v) : R → Rn, g(v, w) : R ×Rn → Rm, and Ω0 ∈ R is a closed convex set. The presence of functional constraints of the form g(v, w) ≤ 0, which couple the parameters and the variables of the problem, is the basic distinction of this statement from the standard one. The coupling constraints make these problems di cult to solve. However, numerous mathematical models contain coupling constraints, which is the reason for the interest in such problems. Methods for solving variational inequalities with coupling constraints have been considered only in few papers, of which we note the paper [1]. On the contrary, there is an extensive literature (e.g., see the review [2]) dealing with the standard statement of problems involving variational inequalities, including solution methods. Problems with coupling constraints arise in numerous elds of mathematics. First, we note economic equilibrium models, which, by de nition, always contain budget constraints implying that the inner product of the price vector by the commodity vector does not exceed some a priori given expenditures. By their very nature, these constraints are always coupling [3]. Generalized statements of n-person games also result in variational inequalities with coupling constraints [4]. Coupling constraints naturally occur in problems of equilibrium programming [5] and hierarchical programming [6]. The development of this direction in applied mathematical physics, where variational inequalities appeared for the rst time, leads to inequalities with coupling constraints [7]. This brief list of problems shows that coupling constraints are not speci c to some particular problem. Quite opposite, they are typical of a wide class of problems. Therefore, it is topical to develop methods for solving problems with coupling constraints. In the present paper we suggest and justify the convergence (asymptotic stability) of a trajectory of a feedback-controlled di erential gradient system to a solution of a variational inequality with coupling constraints.

Two-person game with nash equilibrium in optimal control problems

A two-person game with a Nash equilibrium is formulated for optimal control problems with a free right end and a linear differential system. The game is reduced to the calculation of a fixed point of an extremal mapping, which in turn is reduced to a variational inequality with linear constraints generated by systems of linear differential controllable processes. An extra-gradient iterative method is proposed for calculating the Nash equilibrium of the dynamic game. The convergence of the method is proved.

Extra-proximal methods for solving two-person nonzero-sum games

We consider two-person nonzero-sum game, both in the classical form and in the form of a game with coupled variables. An extra-proximal approach for finding the game’s solutions is suggested and justified. We provide our algorithm with an analysis of its convergence.

Methods of solving systems of convex programming problems

Abstract The problem of matching the interests of the participants in a situation is stated, its different useful interpretations are discussed, and methods of solution are proposed.

Terminal control of boundary models

A terminal optimal control problem for finite-dimensional static boundary models is formulated. The finite-dimensional models determine the initial and terminal states of the plant. The choice of an optimal control drives the plant from one state to another. A saddle-point method is proposed for solving this problem. The convergence of the method in a Hilbert space is proved.

A second-order continuous method for solving quasi-variational inequalities

A continuous method of the gradient type for solving quasi-variational inequalities is examined, and sufficient conditions for this method to converge are found.

Multicriteria equilibrium programming: Extraproximal methods

Multicriteria equilibrium optimization is an efficient tool for mathematical modeling of various situations in operations research, design automation, control, etc. In this paper, a formal formulation of the problem of multicriteria equilibrium optimization is given, and an approach to solving this problem is examined.

A second-order iterative method for solving quasi-variational inequalities

A second-order iterative method for solving quasi-variational inequalities is examined, and sufficient conditions for this method to converge are found.