scholarly journals Two-Step Proximal Method for Equilibrium Problems in Hadamard spaces

2020 ◽  
pp. 71-74
Author(s):  
Yana I. Vedel ◽  
Vladimir V. Semenov ◽  
Kateryna M. Golubeva
Keyword(s):  
Equilibrium Problem ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Saddle Point Problems ◽  
Proximal Method ◽  
Nash Equilibrium Point ◽  
Special Cases ◽  
Step Algorithm ◽  
Monotone Bifunctions

We propose a novel two-step proximal method for solving equilibrium problems in Hadamard spaces. The equilibrium problem is very general in the sense that it includes as special cases many applied mathematical models such as: variational inequalities, optimization problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithm is the analog of the two-step algorithm for solving the equilibrium problem in Hilbert spaces explored earlier. We prove the weak convergence of the sequence generated by the algorithm for pseudo-monotone bifunctions. Our results extend some known results in the literature for pseudo-monotone equilibrium problems.

scholarly journals An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 45
Author(s):  
Wensheng Jia ◽  
Xiaoling Qiu ◽  
Dingtao Peng
Keyword(s):  
Exact Solution ◽  
Bounded Rationality ◽  
Equilibrium Problems ◽  
Approximation Theorem ◽  
Optimization Problems ◽  
Vector Equilibrium Problems ◽  
Special Cases ◽  
Full Rationality ◽  
Vector Equilibrium ◽  
Special Case

In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases.

Properties of the Solution Set for Generalized Equilibrium Problems with Lower and Upper Bounds in FC-Metric Spaces

2013 ◽  
Vol 671-674 ◽  
pp. 1557-1560
Author(s):  
Kai Ting Wen
Keyword(s):  
Metric Spaces ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Upper Bounds ◽  
Solution Set ◽  
Lower And Upper Bounds ◽  
Paper Properties ◽  
Special Cases ◽  
Mathematical Problems ◽  
Generalized Equilibrium

The equilibrium problem includes many fundamental mathematical problems, e.g., optimization problems, saddle point problems, fixed point problems, economics problems, comple- mentarity problems, variational inequality problems, mechanics, engineering, and others as special cases. In this paper, properties of the solution set for generalized equilibrium problems with lower and upper bounds in FC-metric spaces are studied. In noncompact setting, we obtain that the solution set for generalized equilibrium problems with lower and upper bounds is nonempty and compact. Our results improve and generalize some recent results in the reference therein.

scholarly journals General Iterative Methods for Equilibrium Problems and Infinitely Many Strict Pseudocontractions in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Peichao Duan ◽  
Aihong Wang
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Iterative Methods ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Common Element ◽  
Convergence Theorems ◽  
Pseudocontractive Mappings

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of fixed point of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by many others.

scholarly journals Existence and Duality of Generalized ε-Vector Equilibrium Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Hong-Yong Fu ◽  
Bin Dan ◽  
Xiang-Yu Liu
Keyword(s):  
Equilibrium Problem ◽  
Equilibrium Problems ◽  
Existence Theorems ◽  
Vector Equilibrium Problem ◽  
Vector Equilibrium Problems ◽  
Kkm Theorem ◽  
Special Cases ◽  
Primal And Dual Problems ◽  
Primal And Dual ◽  
Vector Equilibrium

We consider a generalized ε-vector equilibrium problem which contain vector equilibrium problems and vector variational inequalities as special cases. By using the KKM theorem, we obtain some existence theorems for the generalized ε-vector equilibrium problem. We also investigate the duality of this generalized ε-vector equilibrium problem and discuss the equivalence relation between solutions of primal and dual problems.

scholarly journals On the existence and stability of solutions to stochastic equilibrium problems

2020 ◽  
Author(s):  
Anh Quoc Lam ◽  
Hai Xuan Nguyen ◽  
Kien Trung Nguyen ◽  
Quan Hong Nguyen ◽  
Dang Thi My Van
Keyword(s):  
Stochastic Optimization ◽  
Equilibrium Problems ◽  
Existence Of Solutions ◽  
Optimization Problems ◽  
Lower Semicontinuous ◽  
Probability Measures ◽  
Stability Of Solutions ◽  
Stochastic Equilibrium ◽  
Special Cases ◽  
Existence And Stability

In this paper we consider stochastic equilibrium problems involving parameter of probability measures. Employing KKM-Fan xed point theorem, conditions for the existence of solutions to such problems are established. We then propose new metric concepts on the underlying stochastic spaces and study some properties corresponding to these metrics. Afterwards, we study sucient conditions for the solution mappings of such problems, that are closed, upper (lower) semicontinuous and continuous with respect to the mentioned metrics. Finally, the special cases of stochastic optimization problems are taken into account as the applications.

scholarly journals Strong Convergence Results of Split Equilibrium Problems and Fixed Point Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Li-Jun Zhu ◽  
Hsun-Chih Kuo ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Split Equilibrium Problem ◽  
Convergence Results

In this paper, we investigate the split equilibrium problem and fixed point problem in Hilbert spaces. We propose an iterative scheme for solving such problem in which the involved equilibrium bifunctions f and g are pseudomonotone and monotone, respectively, and the operators S and T are all pseudocontractive. We show that the suggested scheme converges strongly to a solution of the considered problem.

scholarly journals Schur complement preconditioners for multiple saddle point problems of block tridiagonal form with application to optimization problems

2018 ◽  
Vol 39 (3) ◽  
pp. 1328-1359 ◽  
Author(s):  
Jarle Sogn ◽  
Walter Zulehner
Keyword(s):  
Saddle Point ◽  
Hilbert Spaces ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Schur Complement ◽  
Saddle Point Problems ◽  
Schur Complements ◽  
The Difference ◽  
Optimality Systems ◽  
Tridiagonal Form

Abstract The importance of Schur-complement-based preconditioners is well established for classical saddle point problems in $\mathbb{R}^N \times \mathbb{R}^M$. In this paper we extend these results to multiple saddle point problems in Hilbert spaces $X_1\times X_2 \times \cdots \times X_n$. For such problems with a block tridiagonal Hessian and a well-defined sequence of associated Schur complements, sharp bounds for the condition number of the problem are derived, which do not depend on the involved operators. These bounds can be expressed in terms of the roots of the difference of two Chebyshev polynomials of the second kind. If applied to specific classes of optimal control problems the abstract analysis leads to new existence results as well as to the construction of efficient preconditioners for the associated discretized optimality systems.

scholarly journals ALGORITHM FOR VARIATIONAL INEQUALITY PROBLEM OVER THE SET OF SOLUTIONS THE EQUILIBRIUM PROBLEMS

2020 ◽  
pp. 18-30
Author(s):  
Ya. I. Vedel ◽  
S. V. Denisov ◽  
V. V. Semenov
Keyword(s):  
Variational Inequality ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Proximal Method ◽  
Iterative Regularization ◽  
Inequality Problem ◽  
Lipschitz Continuous ◽  
Convergence Of Sequences ◽  
Continuous Operators ◽  
Monotone Bifunctions

In this paper, we consider bilevel problem: variational inequality problem over the set of solutions the equilibrium problems. To solve this problem, an iterative algorithm is proposed that combines the ideas of a two-stage proximal method and iterative regularization. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz continuous operators, the theorem on strong convergence of sequences generated by the algorithm is proved.

scholarly journals ABOUT THE TWO-STAGE PROXIMAL METHOD FOR SOLVING OF EQUILIBRIUM PROBLEMS

2019 ◽  
pp. 23-31 ◽  
Author(s):  
Ya. I. Vedel ◽  
V. V. Semenov ◽  
L. M. Chabak
Keyword(s):  
Game Theory ◽  
Hilbert Space ◽  
Approximate Solution ◽  
Mathematical Programming ◽  
Weak Convergence ◽  
Equilibrium Problem ◽  
Equilibrium Problems ◽  
Proximal Method ◽  
Existence Of A Solution ◽  
Mathematical Programming Problems

In this paper, the weak convergence of an iterative twostage proximal method for the approximate solution of the equilibrium problem in a Hilbert space is investigated. This method was recently been developed by Vedel and Semenov and can be used to solve mathematical programming problems, variational inequalities and game theory problems. The analysis of the convergence of the method was carried out under the assumption of the existence of a solution of the equilibrium problem and under conditions weaker than the previously considered ones.

scholarly journals Some Results on Strictly Pseudocontractive Nonself-Mappings and Equilibrium Problems in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yan Hao ◽  
Sun Young Cho
Keyword(s):  
Strong Convergence ◽  
Equilibrium Problem ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Common Elements ◽  
Projection Algorithms ◽  
Convergence Theorems

An equilibrium problem and a strictly pseudocontractive nonself-mapping are investigated. Strong convergence theorems of common elements are established based on hybrid projection algorithms in the framework of real Hilbert spaces.

Sign in / Sign up

Export Citation Format

Share Document