有限理性下参数最优化问题解的稳定性

主要研究有限理性下参数最优化问题解的稳定性. 即在两类扰动即目标函数及可行集二者, 目标函数、可行集及参数三者分别同时发生扰动的情形下, 对参数最优化问题引入一个抽象的理性函数, 分别建立了参数最优化问题的有限理性模型M, 运用``通有'的方法, 得到了上述两种扰动情形下相应的有限理性模型M的结构稳定性及对\varepsilon-平衡(解)的鲁棒性, 即有限理性下绝大多数的参数最优化问题的解都 是稳定的, 并以一个例子说明所得的稳定性结果均是正确的.     

多面体集下多目标优化问题近似解的若干性质

研究了多目标优化问题的近似解. 首先证明了多面体集是 co-radiant集,并证明了一些性质. 随后研究了多面体集下多目标优化问题近似解的特殊性质.     

Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems

In this paper we establish sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous. All kinds of semicontinuity are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity and closedness. Moreover, we investigate both the “weak” and “strong” solutions of quasiequilibrium problems.     

Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems

In this article, we study the parametric vector quasi-equilibrium problem (PVQEP). We investigate existence of solution for PVQEP and continuities of the solution mappings of PVQEP. In particular, results concerning the lower semicontinuity of the solution mapping of PVQEP are presented.      

The stability of the solution sets for set optimization problems via improvement sets

ABSTRACT

The aim of this paper is to investigate the stability of the solution sets for set optimization problems via improvement sets. Firstly, we consider the relations among the solution sets for optimization problem with set optimization criterion. Then, the closeness and the convexity of solution sets are discussed. Furthermore, the upper semi-continuity, Hausdorff upper semi-continuity and lower semi-continuity of solution mappings to parametric set optimization problems via improvement sets are established under some suitable conditions. These results extend and develop some recent works in this field.     

参数弱向量平衡问题解集映射的连续性

运用非线性标量化方法, 讨论参数弱向量平衡问题解集映射的上半连续性和下半连续性, 并举例说明了所得结果的正确性.     

Stability for Parametric Vector Quasi-Equilibrium Problems With Variable Cones

This article is devoted to the study of stability conditions for a class of quasi-equilibrium problems with variable cones in normed spaces. We introduce concepts of upper and lower semicontinuity of vector-valued mappings involving variable cones and their properties, we also propose a key hypothesis. Employing this hypothesis and such concepts, we investigate sufficient/necessary conditions of the Hausdorff semicontinuity/continuity for solution mappings to such problems. We also discuss characterizations for the hypothesis which do not contain information regarding solution sets. As an application, we consider the special case of traffic network problems. Our results are new or improve the existing ones.     

An algorithm to solve polyhedral convex set optimization problems

An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain satisfying two conditions: the attainment of the infimum and minimality with respect to a set relation. In the first phase of the algorithm, a linear vector optimization problem, called the vectorial relaxation, is solved. The resulting pre-solution yields the attainment of the infimum but, in general, not minimality. In the second phase of the algorithm, minimality is established by solving certain linear programs in combination with vertex enumeration of some values of the objective map.     

Well-posedness and stability of solutions for set optimization problems

In this paper, some characterizations for the generalized l-B-well-posedness and the generalized u-B-well-posedness of set optimization problems are given. Moreover, the Hausdorff upper semi-continuity of l-minimal solution mapping and u-minimal solution mapping are established by assuming that the set optimization problem is l-H-well-posed and u-H-well-posed, respectively. Finally, the upper semi-continuity and the lower semi-continuity of solution mappings to parametric set optimization problems are investigated under some suitable conditions.     

On the stability of the solution mapping for parametric traffic network problems

In this paper, we introduce the parametric traffic network problems. Afterward, a key hypothesis is introduced by virtue of a parametric gap function to considered problems, and we prove that this hypothesis is not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of the solution mapping for parametric traffic network problems.     

On stability of solutions to parametric generalized affine variational inequalities

This paper deals with the stability of the solution set to parametric generalized affine variational inequalities with constraint set being defined by finitely many convex quadratic functions. The obtained results develop and complement the published ones.     

Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems

In this paper, the lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems without the assumptions of monotonicity and compactness is established by using a new proof method which is different from the ones used in the literature.     

Note: An algorithm to solve polyhedral convex set optimization problems

We provide a counterexample to the remark in Löhne and Schrage [An algorithm to solve polyhedral convex set optimization problems, Optimization 62 (2013), pp. 131-141] that every solution of a polyhedral convex set optimization problem is a pre-solution. A correct statement is that every solution of a polyhedral convex set optimization problem obtained by the algorithm SetOpt is a pre-solution. We also show that every finite infimizer and hence every solution of a polyhedral convex set optimization problem contains a pre-solution.     

Semicontinuity of the solution mappings to weak generalized parametric Ky Fan inequality problems with trifunctions

In this article, we discuss the lower semicontinuity of solution maps without the condition of C-strict monotonicity for two classes of weak generalized parametric Ky Fan inequalities under the case that the f-solution set be a general set-valued one. Our results extend the recent ones in the literature (e.g. Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005); C.R. Chen and S.J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim. 6 (2010), pp. 141–151; Gong, X.H., Yao, J.C.: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 197–205 (2008); Gong, X.H.: Continuity of the solution set to parametric weak vector equilibrium problems. J. Optim. Theory Appl. 139, 35–46 (2008)). Several examples are given for the illustration of our results.     

Stability analysis of the Pareto optimal solutions for some vector boolean optimization problem

In this article we consider the boolean optimization problem of finding the set of Pareto optimal solutions. The vector objectives are the positive cuts of linear functions to the non-negative semi-axis. Initial data are subject to perturbations, measured by the l ₁-norm in the parameter space of the problem. We present the formula expressing the extreme level (stability radius) of such perturbations, for which a particular solution remains Pareto optimal.     

Stability of weak vector variational inequality

In this paper, a key assumption is introduced by virtue of a parametric gap function. Then, by using the key assumption, sufficient conditions of the continuity and Hausdorff continuity of a solution set map for a parametric weak vector variational inequality are obtained in Banach spaces with the objective space being finite-dimensional.     

Well-posedness for parametric optimization problems with variational inclusion constraint

In this paper, we study the well-posedness for the parametric optimization problems with variational inclusion problems as constraint (or the perturbed problem of optimization problems with constraint). Furthermore, we consider the relation between the well-posedness for the parametric optimization problems with variational inclusion problems as constraint and the well-posedness in the generalized sense for variational inclusion problems.     

Well-posedness and stability in vector optimization problems using Henig proper efficiency

In this article, we study well-posedness and stability aspects for vector optimization in terms of minimizing sequences defined using the notion of Henig proper efficiency. We justify the importance of set convergence in the study of well-posedness of vector problems by establishing characterization of well-posedness in terms of upper Hausdorff convergence of a minimizing sequence of sets to the set of Henig proper efficient solutions. Under certain compactness assumptions, a convex vector optimization problem is shown to be well-posed. Finally, the stability of vector optimization is discussed by considering a perturbed problem with the objective function being continuous. By assuming the upper semicontinuity of certain set-valued maps associated with the perturbed problem, we establish the upper semicontinuity of the solution map.