Iterative algorithms for generalized mixed equilibrium problems

In this paper, we consider a generalized mixed equilibrium problem in real Hilbert space. Using the auxiliary principle, we define a class of resolvent mappings. Further, using fixed point and resolvent methods, we give some iterative algorithms for solving generalized mixed equilibrium problem. Furthermore, we prove that the sequences generated by iterative algorithms converge weakly to the solution of generalized mixed equilibrium problem. These results require monotonicity (θ-pseudo monotonicity) and continuity (Lipschitz continuity) for mappings.     

A relaxed extragradient-like method for a generalized mixed equilibrium problem,a general system of generalized equilibria and a fixed point problem

Very recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025–1033] suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a Hilbert space. In this paper, based on Takahashi–Takahashi’s iterative method and well-known extragradient method we introduce a relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a Hilbert space and then obtain a strong convergence theorem. Utilizing this theorem, we establish some new strong convergence results in fixed point problems, variational inequalities, mixed equilibrium problems and systems of generalized equilibria.     

Existence theorems and iterative approximation methods for generalized mixed equilibrium problems for a countable family of nonexpansive mappings

In this paper, we introduce the new generalized mixed equilibrium problem basing on hemicontinuous and relaxed monotonic mapping. Using the KKM technique, we obtain the existence of solutions for the generalized mixed equilibrium problem in a Banach space. Furthermore, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of a countable family of nonexpansive mappings. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.     

A new class of bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in a real Banach space. By using the auxiliary principle technique, new iterative algorithms for solving the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems involving set-valued mappings are suggested and analyzed. Existence of solutions and strong convergence of the iterative sequences generated by the algorithms are proved under quite mild conditions. The behavior of the solution set of the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems is also discussed. These results are new and generalize some recent results in this field.     

New Scalarizing Approach to the Stability Analysis in Parametric Generalized Ky Fan Inequality Problems

This paper gives sufficient conditions for the upper and lower semicontinuities of the solution mapping of a parametric mixed generalized Ky Fan inequality problem. We use a new scalarizing approach quite different from traditional linear scalarization approaches which, in the framework of the stability analysis of solution mappings of equilibrium problems, were useful only for weak vector equilibrium problems and only under some convexity and strict monotonicity assumptions. The main tools of our approach are provided by two generalized versions of the nonlinear scalarization function of Gerstewitz. Our stability results are new and are obtained by a unified technique. An example is given to show that our results can be applied, while some corresponding earlier results cannot.     

Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces

In this paper, a new system of parametric generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings in real Banach spaces is introduced and studied. We first generalize the notion of the Yosida approximation in Hilbert spaces introduced by Moudafi to reflexive Banach spaces. Further, by using the notion of the Yosida approximation, we consider a system of parametric generalized Wiener-Hopf equation problems and show its equivalence to the system of parametric generalized mixed implicit equilibrium problems. By using a fixed point formulation of the system of parametric generalized Wiener-Hopf equation problems, we study the behavior and sensitivity analysis of a solution set of the system of parametric generalized mixed implicit equilibrium problems. We prove that, under suitable assumptions, the solution set of the system of parametric generalized mixed implicit equilibrium problems is nonempty, closed and Lipschitz continuous with respect to the parameters. Our results are new, and improve and generalize some known results in this field.     

THE SPLIT GENERALIZED EQUILIBRIUM PROBLEM INVOLVING INVERSE-STRONGLY MONOTONE MAPPINGS AND ITS CONVERGENCE ALGORITHMS

In this paper, we introduce a split generalized equilibrium problem and consider some iterative sequences to find a solution of the equilibrium problem such that its image under a given bounded linear operator is a solution of another equilibrium problem. We obtain some strong and weak convergence theorems.     

Existence and iterative algorithm of solutions for a class of bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) to compute the approximate solutions of the generalized mixed equilibrium problems (GMEP) and bilevel generalized mixed equilibrium problems (BGMEP) involving set-valued mappings is introduced. By using a minimax inequality, the existence and uniqueness of solutions of the AGMEP is proved under quite mild conditions. By using auxiliary principle technique, new iterative algorithm to compute the approximate solutions of the GMEP and the BGMEP are suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithms are proved under quite mild assumptions. These results are new and generalize some recent results in this field.     

Existence and algorithms for bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) to compute the approximate solutions of the generalized mixed equilibrium problems (GMEP) and bilevel generalized mixed equilibrium problems (BGMEP) involving set-valued mappings is introduced. By using a minimax inequality, the existence and uniqueness of solutions of the AGMEP is proved under quite mild conditions. By using auxiliary principle technique, new iterative algorithm to compute the approximate solutions of the GMEP and the BGMEP is suggested and analyzed. Strong convergence of the iterative sequences generated by the proposed algorithms is proved under quite mild assumptions. These results are new and generalize some recent results in this field.     

与渐近非扩张半群不动点问题相关的分裂等式混合均衡问题及其应用

研究了与渐近非扩张半群不动点问题相关的分裂等式混合均衡问题.在等式约束下,为同时逼近两个空间中混合均衡问题和渐近非扩张半群不动点问题的公共解,借助收缩投影方法引出了一种迭代程序.在适当条件下,该迭代算法的强收敛性被证明.文末还把所得结果应用于分裂等式混合变分不等式问题和分裂等式凸极小化问题.     

On the convergence of cross decomposition

Cross decomposition is a recent method for mixed integer programming problems, exploiting simultaneously both the primal and the dual structure of the problem, thus combining the advantages of Dantzig—Wolfe decomposition and Benders decomposition. Finite convergence of the algorithm equipped with some simple convergence tests has been proved. Stronger convergence tests have been proposed, but not shown to yield finite convergence.In this paper cross decomposition is generalized and applied to linear programming problems, mixed integer programming problems and nonlinear programming problems (with and without linear parts). Using the stronger convergence tests finite exact convergence is shown in the first cases. Unbounded cases are discussed and also included in the convergence tests. The behaviour of the algorithm when parts of the constraint matrix are zero is also discussed. The cross decomposition procedure is generalized (by using generalized Benders decomposition) in order to enable the solution of nonlinear programming problems.     

Banach空间中完全广义混合隐似平衡问题的辅助原理和迭代算法

在Banach空间中利用双线性连续泛函F代替内积引进了新的一类完全广义混合隐似平衡问题，引进了F强单调的概念，提出了该平衡问题的广义辅助问题，证明了广义辅助问题的收敛定理，给出了新的算法和由此算法产生的迭代序列的收敛特征．     

Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces

In this paper, a generalized vector equilibrium problem with set-valued maps defined on a reflexive Banach space is considered. By using the recession method, we first give the conditions under which the solution set is non-empty, convex and weakly compact, and then extend it to the strong generalized vector equilibrium problem. This facilitates generalizing and modifying various existence theorems. Furthermore, the topological properties of the solution set are studied and it is shown that the solution set includes some boundary points.     

反Banach空间内涉及广义混合似臣分不等式问题的双水平广义混合平衡问题

在自反Banach空间内,引入和研究了一类新的涉及广义混合似变分不等式问题的双水平广义混合平衡问题(BGMEP).首先,为了计算BGMEP的近似解,引入了一类辅助广义混合平衡问题(AGMEP).由使用一极小极大不等式,在没有任何强制条件的相当温和假设下,证明了AGMEP解的存在性和唯一性.利用辅助原理技巧,建议和分析了一类计算BGMEP的近似解的新迭代算法.在没有任何强制条件的相当温和假设下,证明了由算法生成的迭代序列的强收敛性.这些结果是新的并且推广了这一领域内某些最近结果.     

An extragradient-type method for solving nonmonotone quasi-equilibrium problems

In this paper, a quasi-equilibrium problem with a nonmonotone bifunction is considered in a finite-dimensional space. The primary difficulty with this problem is related to the fact that one must simultaneously solve a nonmonotone equilibrium problem and calculate a fixed point of a multivalued mapping. An extragradient-type method is presented and analysed for its solution. The convergence of the method is proved under the assumption that the solution set of an associated dual equilibrium problem is nonempty. Finally, some numerical experiments are reported.     

Existence and stability of solutions for a class of generalized vector equilibrium problems

In this paper, two existence theorems concerning the strong efficient solutions and the weakly efficient solutions of generalized vector equilibrium problems are derived by using the Fan-KKM Theorem and an existence theorem for the efficient solutions of generalized vector equilibrium problems is established by using the scalarization method. Moreover, the lower semicontinuity of the strong efficient solution mapping and the weakly efficient solution mapping to parametric generalized vector equilibrium problems are showed under suitable conditions with neither monotonicity nor any information of the solution mappings. Finally, some applications to the vector optimization problems and the Stackelberg equilibrium problem are also given.     

The network equilibrium problem with mixed demand

We formulate the network equilibrium problem with mixed demand which generalizes the problems of network equilibrium with fixed and elastic demand. We prove the equilibrium conditions for this problem and propose some conditions of existence of a solution that are based on the coercivity property.We establish a connection between the problem of network equilibrium with mixed demand and the problem of auction equilibrium. The results of test calculations are presented for a model example.     

Banach空间中广义混合平衡问题,变分不等式问题和不动点问题的混杂投影方法

在Banach空间中,一个新的混杂投影迭代程序被引入来逼近广义混合平衡问题解集,变分不等式问题解集和一个相对弱非扩张映射的不动点集的公共元．所得结果改进和推广了最近一些文献的相应结果．     

An algorithm based on the generalized D-gap function for equilibrium problems

The equilibrium problem (EP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. In this paper, we propose an algorithm for minimizing the problem and analyze some convergence properties of the proposed algorithm. Under some reasonable conditions, we show that the iteration sequence generated by the algorithm is globally convergent and converges to a solution to the EP and the generalized D-gap function provides a global error bound for the algorithm.     

Levitin–Polyak Well-Posedness for Optimization Problems with Generalized Equilibrium Constraints

In this paper, we consider Levitin–Polyak well-posedness of parametric generalized equilibrium problems and optimization problems with generalized equilibrium constraints. Some criteria for these types of well-posedness are derived. In particular, under certain conditions, we show that generalized Levitin–Polyak well-posedness of a parametric generalized equilibrium problem is equivalent to the nonemptiness and compactness of its solution set. Finally, for an optimization problem with generalized equilibrium constraints, we also obtain that, under certain conditions, Levitin–Polyak well-posedness in the generalized sense is equivalent to the nonemptiness and compactness of its solution set.