L. A. Artem’eva

Extraproximal method for solving two-person saddle-point games

An equilibrium model is proposed for a two-person saddle-point game with partially coincident or conflicting interests. Meaningful interpretations of such a game are discussed. Three variants of the extraproximal method for finding an equilibrium point are proposed, and their convergence is proved.

Multicriteria equilibrium programming: Extragradient method

Multicriteria equilibrium programming includes as its particular cases mathematical programming, saddle point calculation, the multicriteria search for Pareto solutions, minimization with an equilibrium choice of the feasible set, etc. An extragradient method is proposed for the numerical solution of the multicriteria equilibrium programming problem, and the convergence of this method is examined.

Regularized extragradient method for solving parametric multicriteria equilibrium programming problem

A regularized extragradient method is designed for solving unstable multicriteria equilibrium programming problems. The convergence of the method is investigated, and a regularizing operator is constructed.

Continuous extragradient method for a parametric multicriteria equilibrium programming problem

We consider a multicriteria equilibrium programming problem including, as special cases, the mathematical programming problem, the problem of finding a saddle point, the multicriteria problem of finding a Pareto point, the minimization problem with an equilibrium choice of an admissible set, etc. We suggest a continuous version of the extragradient method with prediction and analyze its convergence.

Regularized extragradient method for searching for an equilibrium point in two-person saddle-point games

A two-person saddle-point game with approximately given input data is examined. Since, in games of this type, the search for an equilibrium point is unstable with respect to perturbations in the input data, two variants of the regularized extragradient method are proposed. Their convergence is analyzed, and a regularizing operator is constructed.

Differential extragradient method for finding an equilibrium in two-person saddle-point games

We describe an equilibrium model of a two-person saddle-point game with partially opposite or coinciding interests. To find an equilibrium point, we suggest an extraproximal method in the form of a Cauchy problem for a system of ordinary differential equations with prediction. We consider three versions of this method and analyze their convergence.

Regularized continuous extragradient method for a multicriterial equilibrium programming problem

For solving unstable multicriterial problems, we suggest a regularized version of the continuous extragradient method, analyze its convergence, and construct a regularizing operator.

Extragradient method for finding a saddle point in a multicriteria problem with dynamics

We consider an optimal control problem for a linear system of ordinary differential equations with an implicitly given boundary condition connected with a multicriteria problem. Such problems arise, for example, in the study of controlled objects that lose their stability under the influence of external perturbations, where it is required to return an object to stability by means of an appropriate choice of the control. We describe a possible mathematical model of this kind, propose an extragradient method for recovering the stability, and investigate its convergence.

Extragradient method for solving an optimal control problem with implicitly specified boundary conditions

An optimal control problem formulated as a system of linear ordinary differential equations with boundary conditions implicitly specified as a solution to a finite-dimensional minimization problem is considered. An extragradient method for solving this problem is proposed, and its convergence is studied.

Regularized extragradient method in multicriteria control problems with inaccurate data

The class of Hilbert space multicriteria optimization problems considered in the paper includes control problems for various dynamical systems with lumped as well as distributed parameters. An equilibrium point is sought under the assumption that the criteria and their derivatives are known approximately. We use a regularized extragradient method and prove its convergence. As a sample application of the general theory, we consider a control problem for a parabolic equation with two criteria.