Lower Semicontinuous Regularization for Vector-Valued Mappings

The paper is devoted to studying the lower semicontinuity of vector-valued mappings. The main object under consideration is the lower limit. We first introduce a new definition of an adequate concept of lower and upper level sets and establish some of their topological and geometrical properties. A characterization of semicontinuity for vector-valued mappings is thereafter presented. Then, we define a concept of vector lower limit, proving its lower semicontinuity, and furnishing in this way a concept of lower semicontinuous regularization for mappings taking their values in a complete lattice. The results obtained in the present work subsume the standard ones when the target space is finite dimensional. In particular, we recapture the scalar case with a new flexible proof. In addition, extensions of usual operations of lower and upper limits for vector-valued mappings are explored. The main result is finally applied to obtain a continuous D.C. decomposition of continuous D.C. mappings. Dedicated to Alex Rubinov in honor of his 65th birthday     

Characterizations and Applications of Prequasi-Invex Functions

In this paper, two new types of generalized convex functions are introduced. They are called strictly prequasi-invex functions and semistrictly prequasi-invex functions. Note that prequasi-invexity does not imply semistrict prequasi-invexity. The characterization of prequasi-invex functions is established under the condition of lower semicontinuity, upper semicontinuity, and semistrict prequasi-invexity, respectively. Furthermore, the characterization of semistrictly prequasi-invex functions is also obtained under the condition of prequasi-invexity and lower semicontinuity, respectively. A similar result is also obtained for strictly prequasi-invex functions. It is worth noting that these characterizations reveal various interesting relationships among prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions. Finally, prequasi-invex, semistrictly prequasi-invex, and strictly prequasi-invex functions are used in the study of optimization problems.     

Continuity of the efficient solution mapping for vector optimization problems

This paper aims at investigating the continuity of the efficient solution mapping of perturbed vector optimization problems. First, we introduce the concept of the level mapping. We give sufficient conditions for the upper semicontinuity and the lower semicontinuity of the level mapping. The upper semicontinuity and the lower semicontinuity of the efficient solution mapping are established by using the continuity properties of the level mapping. We establish a corollary about the lower semicontinuity of the minimal point set-valued mapping. Meanwhile, we give some examples to illustrate that the corollary is different from the ones in the literature.     

Semicontinuity of the Approximate Solution Sets of Multivalued Quasiequilibrium Problems

We consider two kinds of approximate solutions and approximate solution sets to multivalued quasiequilibrium problems. Sufficient conditions for the lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness of these approximate solution sets are established. Applications in approximate quasivariational inequalities, approximate fixed points, and approximate quasioptimization problems are provided.     

On the Stability of the Solution Sets of General Multivalued Vector Quasiequilibrium Problems

We give sufficient conditions for the semicontinuity of solution sets of general multivalued vector quasiequilibrium problems. All kinds of semicontinuities are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness. Moreover, we investigate the weak, middle, and strong solutions of quasiequilibrium problems. Many examples are provided to give more insights and comparisons with recent existing results.     

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.     

Semicontinuity Results on Parametric Vector Variational Inequalities with Polyhedral Constraint Sets

New results on the lower semicontinuity and upper semicontinuity properties of the Pareto solutions to a parametric vector variational inequality with a polyhedral constraint set are obtained by scalarization approaches.     

Continuity of the optimal value function in indefinite quadratic programming

This paper characterizes the continuity property of the optimal value function in a general parametric quadratic programming problem with linear constraints. The lower semicontinuity and upper semicontinuity properties of the optimal value function are studied as well.     

Upper and Lower Semicontinuity for Set-Valued Mappings Involving Constraints

In vector optimization, several authors have studied the upper and lower semicontinuity for mappings involving constraints in topological vector spaces partially ordered through a cone with nonempty interior. In this paper, we give conditions about the upper and lower semicontinuity in the case that the ordering cone in the parameter space has possibly empty interior, as it happens in many function spaces and seqence spaces.     

Stability analysis for vector quasiequilibrium problems

This paper is devoted to the continuity of the solution mapping for vector quasiequilibrium problems under mapping perturbations. We show that the solution mapping is upper semicontinuous and Hausdorff upper semicontinuous. Sufficient conditions for the lower semicontinuity and Hausdorff lower semicontinuity of the solution mapping are established. Finally, we apply our results to traffic network problems as example.     

Stability of Parametric Quasivariational Inequality of the Minty Type

In this paper, stability of a parametric quasivariational inequality of the Minty type is studied via various sufficient conditions characterizing upper and lower semicontinuity of the solution sets as well as the approximate solution sets. Sufficient conditions ensuring upper semicontinuity of the approximate solution sets of an optimization problem with quasivariational inequality constraints are also presented.     

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.     

Gap Continuity of Multimaps

We introduce a notion of continuity for multimaps (or set-valued maps) which is mild. It encompasses both lower semicontinuity and upper semicontinuity. We give characterizations and we consider some permanence properties. This notion can be used for various purposes. In particular, it is used for continuity properties of subdifferentials and of value functions in parametrized optimization problems. We also prove an approximate selection theorem.     

Continuity of the solution mappings to parametric generalized strong vector equilibrium problems

In this paper, we establish the upper semicontinuity and lower semicontinuity of solution mappings to a parametric generalized strong vector equilibrium problem with setvalued mappings by using a scalarization method and a density result. The results improve the corresponding ones in the literature. Some examples are given to illustrate our results.     

Epi and coepi-analysis of one class of vector-valued mappings

This article deals with a new characterization of lower semicontinuity of vector-valued mappings in normed spaces. We study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their epigraphs and coepigraphs, respectively. We show that if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to the procedure of their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower semicontinuous. Using these results we propose some regularization schemes for vector-valued functions. In the case when there are no assumptions on the topological interior of the ordering cone, we introduce a new concept of lower semicontinuity for vector-valued mappings, the so-called epi-lower semicontinuity, which is closely related to the closedness of epigraphs of such mappings, and study their main properties. All principal notions and assertions are illustrated by numerous examples.     

Semicontinuity and Quasiconvex Functions

Criteria are derived for quasiconvex functions under lower semicontinuity and upper semicontinuity conditions. The results thus obtained generalize earlier results for convex functions. We also give new conditions under which a given function is r-convex in the sense given by Avriel.     

Image of a parametric optimization problem and continuity of the perturbation function

In this note, we consider the notion of the image of a parametric optimization problem and show that the lower semicontinuity and upper semicontinuity properties of its marginal function can be equivalently expressed as two geometric relations in the image space. These results generalize some existing statements in the literature.     

Gap Continuity of Multimaps

We introduce a notion of continuity for multimaps (or set-valued maps) which is mild. It encompasses both lower semicontinuity and upper semicontinuity. We give characterizations and we consider some permanence properties. This notion can be used for various purposes. In particular, it is used for continuity properties of subdifferentials and of value functions in parametrized optimization problems. We also prove an approximate selection theorem.      

To the problem of continuity of many-valued mappings

We study the problem of the upper and lower semicontinuity of the union and intersection for a family of many-valued mappings. We establish new conditions of lower semicontinuity for the intersection of a family of lower semicontinuous mappings.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1519–1525, November, 1995.     

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.