On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

Generalized Nash equilibrium problems

The Generalized Nash equilibrium problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields. In this survey paper we aim at discussing its main properties and solution algorithms, pointing out what could be useful topics for future research in the field. The work of Christain Kanzow has been partially supported by the program “Identification, Optimization and Control with Applications in Modern Technologies” of the Elite Network of Bavaria, Germany.     

Existence results for strong vector equilibrium problems and their applications

New existence results for the strong vector equilibrium problem are presented, relying on a well-known separation theorem in infinite-dimensional spaces. The main results are applied to strong cone saddle-points and strong vector variational inequalities providing new existence results, and furthermore they allow recovery of an earlier result from the literature.     

The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions

We extend the Tikhonov regularization method widely used in optimization and monotone variational inequality studies to equilibrium problems. It is shown that the convergence results obtained from the monotone variational inequality remain valid for the monotone equilibrium problem. For pseudomonotone equilibrium problems, the Tikhonov regularized subproblems have a unique solution only in the limit, but any Tikhonov trajectory tends to the solution of the original problem, which is the unique solution of the strongly monotone equilibrium problem defined on the basis of the regularization bifunction.     

Generalized Invex Monotonicity and Its Role in Solving Variational-Like Inequalities

The paper stresses the role of new classes of generalized invex monotonicity in the convergence of iterative schemes for solving a variational-like inequality problem on a closed convex set. This work was supported by Grant NSFC 70432001.     

Generalized Nash equilibrium problem, variational inequality and quasiconvexity

It is well known that the generalized Nash equilibrium problem, a model for multi-leader–follower games, can be reformulated as a quasivariational inequality. We show that, in fact, a reformulation in terms of a variational inequality can be obtained in the general setting of quasiconvex nondifferentiable decision functions. An existence result is deduced.     

Recession methods for generalized vector equilibrium problems

By using the recession method, we give some necessary and/or sufficient condition of solutions of generalized vector equilibrium problems.     

Well-posedness for vector equilibrium problems

We introduce and study two notions of well-posedness for vector equilibrium problems in topological vector spaces; they arise from the well-posedness concepts previously introduced by the same authors in the scalar case, and provide an extension of similar definitions for vector optimization problems. The first notion is linked to the behaviour of suitable maximizing sequences, while the second one is defined in terms of Hausdorff convergence of the map of approximate solutions. In this paper we compare them, and, in a concave setting, we give sufficient conditions on the data in order to guarantee well-posedness. Our results extend similar results established for vector optimization problems known in the literature.      

Gap functions for generalized vector equilibrium problems via conjugate duality and applications

In this article, gap functions for a generalized vector equilibrium problem (GVEP) with explicit constraints are investigated. Under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are investigated by considering the different perturbations to GVEP. Then, gap functions for GVEP are established by using the weak and strong duality results. As application, the proposed approach is applied to construct gap functions for a vector optimization problem and a generalized vector variational inequality problem.     

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2^Y a point-to-set mapping such that for any χ ε K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ^* ε K such that f g(χ^*), y) -int C(χ) for all y ε K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.     

Existence theorems via duality for equilibrium problems with trifunctions

We present an existence result for an equilibrium problem formulated with trifunctions, which is motivated by variational inequalities governed by quasimonotone operators. To prove the existence result, we define the dual problem, and some monotonicity notions for trifunctions. From the main result follow, among others, the Browder–Minty theorem for variational inequalities and Ky Fan’s Minimax theorem. Some applications for mixed equilibrium problems and variational inequalities are given.     

An improved two-step method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is a noncooperative game in which the strategy set of each player, as well as his payoff function, depend on the rival players strategies. As a generalization of the standard Nash equilibrium problem (NEP), the GNEP has recently drawn much attention due to its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. In this paper, we propose an improved two-step (a prediction step and a correction step) method for solving the quasi-variational inequality (QVI) formulation of the GNEP. Per iteration, we first do a projection onto the feasible set defined by the current iterate (prediction) to get a trial point; then, we perform another projection step (correction) to obtain the new iterate. Under certain assumptions, we prove the global convergence of the new algorithm. We also present some numerical results to illustrate the ability of our method, which indicate that our method outperforms the most recent projection-like methods of Zhang et al. (2010).     

Tykhonov well-posedness for lexicographic equilibrium problems

In this paper, we consider the vector equilibrium problems involving lexicographic cone in Banach spaces. We introduce the new concepts of the Tykhonov well-posedness for such problems. The corresponding concepts of the Tykhonov well-posedness in the generalized sense are also proposed and studied. Some metric characterizations of well-posedness for such problems are given. As an application of the main results, several results on well-posedness for the class of lexicographic variational inequalities are derived.     

A new projection algorithm for solving constrained equilibrium problems in Hilbert spaces

ABSTRACT

In this paper, we introduce a new iterative scheme by combining the hyperplane projection method and the inertial technique for constrained equilibrium problems in real Hilbert spaces. The convergence of the proposed algorithm is established without requiring strict paramonotonicity property. The results presented in the paper extend and improve some recent results in the literature. In addition, a numerical example is given to illustrate the efficiency and performance of the proposed method.     

On implicit vector equilibrium problems

In this paper, we introduce and study a class of implicit vector equilibrium problems, which includes a number of (scalar) implicit equilibrium problems, implicit variational inequalities, and implicit complementarity problems as special cases. By using KKM-Fan theorem, we prove some new existence theorems of solutions for this kind of implicit vector equilibrium problems in Hausdorff topological vector spaces. Our results extend and unify some corresponding results of several authors.     

On the proximal point method for equilibrium problems in Hilbert spaces

We analyse a proximal point method for equilibrium problems in Hilbert spaces, improving upon previously known convergence results. We prove global weak convergence of the generated sequence to a solution of the problem, assuming existence of solutions and rather mild monotonicity properties of the bifunction which defines the equilibrium problem, and we establish existence of solutions of the proximal subproblems. We also present a new reformulation of equilibrium problems as variational inequalities ones.     

On the Stability of Generalized Vector Quasivariational Inequality Problems

In this paper, we obtain some stability results for generalized vector quasivariational inequality problems. We prove that the solution set is a closed set and establish the upper semicontinuity property of the solution set for perturbed generalized vector quasivariational inequality problems. These results extend those obtained in Ref. 1. We obtain also the lower semicontinuity property of the solution set for perturbed classical variational inequalities. Several examples are given for the illustration of our results.     

惩罚框架下求解广义Nash均衡问题的分解算法

广义Nash均衡问题(GNEP),是非合作博弈论中一类重要的问题,它在经济学、管理科学和交通规划等领域有着广泛的应用.本文主要提出一种新的惩罚算法来求解一般的广义Nash均衡问题,并根据罚函数的特殊结构,采用交替方向法求解子问题.在一定的条件下,本文证明新算法的全局收敛性.多个数值例子的试验结果表明算法是可行的,并且是有效的.     

A class of filter problems,I: Existence theory

A class of filter problems in which the autocorrelation function of the noise is assumed to be expressible as a convolution is related to a variational problem. The variational problem involves the minimization of the square of theL ₁[0, ] norm of a function y plus the square of theL ₂[0, ] norm of a functionG. The functions y andG are related by a renewal equation. The problems formulated here include as special cases previous problems solved by the authors and their students. In the present paper, the existence and uniqueness of the solution of the variational problems are established.Dedicated to R. BellmanThe research of the first author was supported by NSF Grants Nos. MCS-78-01106 and MCS-75-67947.