On a generalized vector equilibrium problem with bounds

We prove the existence of a solution to the generalized vector equilibrium problem with bounds. We show that several known theorems from the literature can be considered as particular cases of our results, and we provide examples of applications related to best approximations in normed spaces and variational inequalities.     

On vector equilibrium problem

This paper presents some existence results of a vector equilibrium problem. The several important special cases of the vector equilibrium problem are also discussed.     

Random matrix model with external source and a constrained vector equilibrium problem

We consider the random matrix model with external source in the case where the potential V(x) is an even polynomial and the external source has two eigenvalues ±a of equal multiplicity. We show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures (μ₁, μ₂) that solve a constrained vector equilibrium problem. The proof is based on the steepest‐descent analysis of the associated Riemann‐Hilbert problem for multiple orthogonal polynomials. We illustrate our results in detail for the case of a quartic double‐well potential $V(x) = {1 \over 4} x^4 - {t \over 2} x^2$ . We are able to determine the precise location of the phase transitions in the ta‐plane, where either the constraint becomes active or the two intervals in the support come together (or both). © 2010 Wiley Periodicals, Inc.     

Riemann-Hilbert problem associated with Angelesco systems

Angelesco systems of measures with Jacobi-type weights are considered. For such systems, strong asymptotics for the related multiple orthogonal polynomials are found as well as the Szeg?-type functions. In the procedure, an approach from the Riemann-Hilbert problem plays a fundamental role.     

On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem

In this paper, under new assumptions, which do not contain any information about the solution set, the lower semicontinuity of solution mappings to a parametric generalized strong vector equilibrium problem are established by using a scalarization method. These results extend and generalize the corresponding ones in Gong and Yao (J Optim Theory Appl 138:197–205, 2008), Chen and Li (Pac J Optim 6:141–151, 2010) and Li et al. (2012, submitted). Some examples are given to illustrate our results.     

A method of duality for a mixed vector equilibrium problem

In this paper, a dual scheme for a mixed vector equilibrium problem is introduced by using the method of Fenchel conjugate function. Under the stabilization condition, the relationships between the solutions of mixed vector equilibrium problem (MVEP) and dual mixed vector equilibrium problem (DMVEP) are discussed. Moreover, under the same condition, the solutions of MVEP and DMVEP are proved relating to the saddle points of an associated Lagrangian mapping. As applications, this dual scheme is applied to vector convex optimization and vector variational inequality.     

On the generalized vector equilibrium problems

In this paper, generalized vector equilibrium problems are studied and some existence theorems of solutions for these problems in the setting of topological vector spaces are proved. Sufficient conditions for the set of solutions to be compact and convex are given. Our results improve some recent results in this field.     

An existence result for the generalized vector equilibrium problem

In this paper, we prove an existence result for the generalized vector equilibrium problem by using Fan-Browder type fixed-point theorem due to [1].     

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.     

On dual vector equilibrium problems

In this work, we consider the dual vector equilibrium problems in the topological vector spaces setting for a moving cone. We investigate the relationship between solutions of the vector equilibrium problems and those for their perturbations. Our result may be viewed as a refinement and improvement of the paper [L. Huang, Existence of Solutions on Weak Vector Equilibrium Problems, vol. 65, 2006, pp. 795–801.].     

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.     

A remark on a standard and linear vector network equilibrium problem with capacity constraints

In this paper, (weak) vector equilibrium principle with capacity constraints is introduced. A necessary condition that a vector minimum cost flow is a vector equilibrium flow with capacity constraints is obtained. When the number of paths connecting with each pair of source and sink is less than or equal to 2, a sufficient condition for a vector minimum cost flow to be a vector equilibrium flow is also obtained. A generalized (weak) vector equilibrium principle is also introduced. Without any additional assumption, a necessary and sufficient condition for a (weak) vector minimum cost flow to be a generalized (weak) vector equilibrium flow is obtained.     

Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem

This paper deals with the stability for a parametric generalized strong vector equilibrium problem. Under new assumptions, which do not contain any information about the solution set and monotonicity, we establish the lower semicontinuity and upper semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem by using a scalarization method and a density result. These results are improve the corresponding ones in recent literature. Some examples are given to illustrate our results.     

On the matrix permutation problem

This note deals with the problem of permuting elements within columns of a real matrix so as to minimize a real-valued function of row sums. The special case dealing with minimization of maximum row sum has been studied by several authors [1, 4, 6, 7] recently. Here we are concerned primarily with the case in which the matrix has two columns only and the function is Schur-convex.

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Zusammenfassung Diese Note behandelt das Problem, die Spaltenelemente einer reellen Matrix so zu permutieren, daß eine reellwertige Funktion der Zeilensummen minimiert wird. Der Spezialfall der Minimierung der maximalen Zeilensumme wurde jüngst von mehreren Autoren [1, 4, 6, 7] untersucht. Hier betrachten wir primär den Fall, daß die Matrix nur zwei Spalten hat und die Zielfunktion Schur-konvex ist.

     

A projection iterative algorithm for strong vector equilibrium problem

In this paper, iterative algorithm for strong vector equilibrium problem (SVEP) is studied. Firstly, an auxiliary problem for SVEP is introduced and the relationships between these two problems are discussed. Then, based on the auxiliary problem, a projection iterative algorithm for SVEP is proposed. Moreover, analysis of convergence of this iterative algorithm is investigated under suitable conditions of continuity and convexity. The main result obtained in this paper generalizes and improves the corresponding ones of Iusem and Sosa [Iusem AN, Sosa W. Iterative algorithms for equilibrium problems. Optimization. 2003;52(3):301–316.] and Cheng and Liu [Cheng B, Liu SY. An iterative algorithm for vector equilibrium problems. J. Lanzhou Univ. (Nat. Sci.). 2009;45(5):105–109.].     

An iterative method for solving the strong vector equilibrium problem with variable domination structure

In this paper, an iterative algorithm for solving the strong vector equilibrium problem with variable domination structure (VSVEP) is considered. First, an auxiliary problem for the VSVEP is introduced and the relationships between the auxiliary problem and VSVEP are discussed. Then, using the auxiliary principle technique, a projection iterative algorithm to compute the approximate solutions of the VSVEP is proposed and analysed. Furthermore, convergence of the iterative sequences generated by this algorithm is investigated under suitable conditions of continuity and convexity. These results extend and improve some recent works in this field.     

On solving a nonlinear matrix eigenvalue problem

Journal of Mathematical Sciences -