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.     

On supremal functionals defined in <Emphasis Type="Italic">BV</Emphasis>

This paper is devoted to the study of the lower semicontinuity properties of suitable supremal functionals defined on the space of the functions of bounded variation. The main tool in the proofs is a lower semicontinuity theorem for supremal functionals defined on measures that is inspired to the classical Reshetnyak theorem in which integral functionals are involved.      

Lower Semicontinuity Results in Parametric Multivalued Weak Vector Equilibrium Problems and Applications

This article gives new sufficient conditions for the lower semicontinuity of the solution mapping of a parametric multivalued weak vector equilibrium problem with moving cones. A scalarizing approach, based on the signed distance function of Hiriart Urruty is used to discuss this lower semicontinuity property. The main results of the article are obtained under some assumptions different from those introduced earlier by previous linear and nonlinear scalarizing approaches. Some applications to the study of connectedness of weak solution sets of multivalued vector equilibrium problems are given.     

Outer semicontinuity of subdifferential and normal cone mappings

In finite dimensions, the outer semicontinuity of a set-valued mapping is equivalent to the closedness of its graph. In this article, we study the outer semicontinuity of set-valued mappings in connection with their convexifications and linearizations in finite and infinite dimensions. The results are specified to the case where the mappings involved are given by subdifferentials of extended real-valued functions or normal cones to sets. Our developments are important for applications to second-order calculus in variational analysis in which the outer semicontinuity plays a crucial role.     

On the L 1-lower semicontinuity of certain convex functionals defined on BV-vector valued functions

The aim of this work is to provide an L ¹-lower semicontinuity result for an integral convex functional defined on BV-vector valued functions, without any coerciveness and any continuity assumption on the integrand f(x, ξ) with respect to the variable x.      

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.     

On the lower semicontinuity of optimal solution sets

We study the lower semicontinuity of the optimal solution set of a parametric optimization problem. Our results sharpen the main results of Zhao (1997, The lower semicontinuity of optimal solution sets. Journal of Mathematical Analysis and Applications, 207, 240–254. Ref. ). Namely, it is shown that the conclusion of Theorem 1 of is still valid under weaker assumptions, and the conditions on “ε-nontriviality” and uniform continuity of the objective function in Theorems 2 and 3 of can be omitted.     

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.     

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 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.