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.     

A stabilization method for solution of a parametric multicriteria equilibrium programming problem

A multicriteria equilibrium programming problem comprising a mathematical programming problem as a particular case, a multicriteria Pareto-point search problem, a minimization problem with equilibrium selection of the feasible set, etc., is considered. It is assumed that the initial data are known only approximately. In view of the fact that the considered problem is generally unstable with respect to the input data, a regularization method, which is a generalization of the Tikhonov stabilization method, is proposed. Conditions for matching the method parameters to the error in the input data are presented. The convergence of this method is analyzed.     

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.     

Interior proximal extragradient method for equilibrium problems

The Bregman function-based Proximal Point Algorithm (BPPA) is an efficient tool for solving equilibrium problems and fixed-point problems. Extending rather classical proximal regularization methods, the main additional feature consists in an application of zone coercive regularizations. The latter allows to treat the generated subproblems as unconstrained ones, albeit with a certain precaution in numerical experiments. However, compared to the (classical) Proximal Point Algorithm for equilibrium problems, convergence results require additional assumptions which may be seen as the price to pay for unconstrained subproblems. Unfortunately, they are quite demanding – for instance, as they imply a sort of unique solvability of the given problem. The main purpose of this paper is to develop a modification of the BPPA, involving an additional extragradient step with adaptive (and explicitly given) stepsize. We prove that this extragradient step allows to leave out any of the additional assumptions mentioned above. Hence, though still of interior proximal type, the suggested method is applicable to an essentially larger class of equilibrium problems, especially including non-uniquely solvable ones.     

The interior proximal extragradient method for solving equilibrium problems

In this article we present a new and efficient method for solving equilibrium problems on polyhedra. The method is based on an interior-quadratic proximal term which replaces the usual quadratic proximal term. This leads to an interior proximal type algorithm. Each iteration consists in a prediction step followed by a correction step as in the extragradient method. In a first algorithm each of these steps is obtained by solving an unconstrained minimization problem, while in a second algorithm the correction step is replaced by an Armijo-backtracking linesearch followed by an hyperplane projection step. We prove that our algorithms are convergent under mild assumptions: pseudomonotonicity for the two algorithms and a Lipschitz property for the first one. Finally we present some numerical experiments to illustrate the behavior of the proposed algorithms.     

An extragradient method for equilibrium problems on Hadamard manifolds

In this paper, we propose an extragradient algorithm for solving equilibrium problems on Hadamard manifolds to the case where the equilibrium bifunction is not necessarily pseudomonotone. Under mild assumptions, we establish global convergence results. We show that the multiobjective optimization problem satisfies all the hypotheses of our result of convergence, when formulated as an equilibrium problem.     

A problem expanding parametric programming method for solving the job shop scheduling problem

A new zero-one integer programming model for the job shop scheduling problem with minimum makespan criterion is presented. The algorithm consists of two parts: (a) a branch and bound parametric linear programming code for solving the job shop problem with fixed completion time; (b) a problem expanding algorithm for finding the optimal completion time. Computational experience for problems having up to thirty-six operations is presented. The largest problem solved was limited by memory space, not computation time. Efforts are under way to improve the efficiency of the algorithm and to reduce its memory requirements.This report was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract No. N00014-82-K-0329 NR 047-048 with the U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

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.     

Convex programming and the lexicographic multicriteria problem

Mathematical programming formulation of the convex lexicographic multi-criteria problems typically lacks a constraint qualification. Therefore the classical Kuhn-tucker theory fails to characterize their optimal solutions. Furthermore, numerical methods for solving the lexicographic problems are virtually nonexistent. This paper shows that using a recent theory of convex programming, which is free of a constraint qualification assumption, it is possible both to characterize and to calculate the optimal solutions of the convex lexicographic problem.     

New extragradient method for a class of equilibrium problems in Hilbert spaces

The paper proposes a new extragradient algorithm for solving strongly pseudomonotone equilibrium problems which satisfy a Lipschitz-type condition recently introduced by Mastroeni in auxiliary problem principle. The main novelty of the paper is that the algorithm generates the strongly convergent sequences in Hilbert spaces without the prior knowledge of Lipschitz-type constants and any hybrid method. Several numerical experiments on a test problem are also presented to illustrate the convergence of the algorithm.     

Exact solution to a parametric linear programming problem

In this paper, we present a cubically convergent method for finding the largest eigenvalue of a nonnegative irreducible tensor. A cubically convergent method is used to solve an equivalent system of nonlinear equations which is transformed by the tensor eigenvalue problem. Due to particular structure of tensor, Chebyshev’s direction is added to the method with a few extra computation. Two rules are designed such that the descendant property of the search directions is ensured. The global convergence is proved by using the line search technique. Numerical results indicate that the proposed method is competitive and efficient on some test problems.     

An extragradient method for relaxed cocoercive variational inequality and equilibrium problems

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansivemappings, the set of an equilibrium problem and the set of solutions of the variational inequality problem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao, Takahashi and many others.     

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

We consider a saddle-point game of two persons with partly opposite or coinciding interests. For finding an equilibrium, we suggest an extragradient method in the form of the Cauchy problem for a system of ordinary differential equations with feedback. We consider three versions of this method and analyze their convergence.     

A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance

In this paper, using a hybrid extragradient method, we introduce a new iterative process for approximating a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. For this purpose, we introduce Bregman–Lipschitz-type condition for a pseudomonotone bifunction. It seems that these results for pseudomonotone bifunctions are first in reflexive Banach spaces. This paper concludes with certain applications, where we utilize our results to study the determination of a common point of the solution set of a variational inequality problem and the fixed point set of a finite family of multi-valued relatively nonexpansive mappings. A numerical example to support our main theorem will be exhibited.     

Modified extragradient algorithms for solving equilibrium problems

ABSTRACT

In this paper, we introduce some new algorithms for solving the equilibrium problem in a Hilbert space which are constructed around the proximal-like mapping and inertial effect. Also, some convergence theorems of the algorithms are established under mild conditions. Finally, several experiments are performed to show the computational efficiency and the advantage of the proposed algorithm over other well-known algorithms.     

Regularized extragradient method for finding a saddle point in an optimal control problem

We study n-dimensional cubical pseudomanifolds and their cellular mappings. In particular, we consider a discrete n-cube and all of its (n ? 1)-faces. Then, there exist either one or two or four faces of the cube each of which is mapped onto one face.     

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.     

A modified subgradient extragradient method for solving the variational inequality problem

The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. (J. Optim. Theory Appl. 148, 318–335, 2011), replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors proposed extensions and modifications with applications to various problems. In this paper, we introduce a modified subgradient extragradient method by improving the stepsize of its second step. Convergence of the proposed method is proved under standard and mild conditions and primary numerical experiments illustrate the performance and advantage of this new subgradient extragradient variant.