Restricted Weak Upper Semi-continuity of Subdifferentials of Convex Functions on Banach Spaces

Let X be a Banach space and f a continuous convex function on X. Suppose that for each x ∈ X and each weak neighborhood V of zero in X ^* there exists δ > 0 such that $$\partial f(y)\subset\partial f(x)+V\;\;{\rm for\;all}\;y\in X\;{\rm with}\;\|y-x\|<\delta. $$ Then every continuous convex function g with $g \leqslant f$ on X is generically Fréchet differentiable. If, in addition, $\lim\limits_{\|x\|\rightarrow\infty}f(x)=\infty$ , then X is an Asplund space.     

Duality and Weak Efficiency in Vector Variational Problems

We establish weak, strong, and converse duality results for weakly efficient solutions in vector or multiobjective variational problems, which extend and improve recent papers. For this purpose, we consider Kuhn–Tucker optimality conditions, weighting variational problems, and some classes of generalized convex functions, recently introduced, which are extended in this work. Furthermore, a related open question is discussed.     

弱向量均衡问题的含参适定性

本文在实 Banach 空间中研究了弱向量均衡问题的两种适定性.给出了该问题唯一适定与适定的距离刻划.在适当条件下证明了弱向量均衡问题的唯一适定性等价于解的存在性与唯一性.最后, 文章在有限维空间给出了弱向量均衡问题适定的充分性条件.     

含参弱向量均衡问题的适定性

在实Hausdorff拓扑线性空间中研究了含参弱向量均衡问题的适定性.证明了在适当条件下由近似网定义的含参适定性等价于近似解映射的上半连续性,并给出了所研究问题各种适定的充分性条件.     

含参集值弱向量平衡问题解集映射的半连续性

在线性度量空间中,运用标量化技巧在没有单调性与C-凹性假设下得到了含参集值弱向量平衡问题解集映射上/下半连续性的充分性条件.所得结果改进了已有文献的相应结果,并通过例子验证了所得结果.     

Monotone Approximations of Minimum and Maximum Functions and Multi-objective Problems

In this paper the problem of accomplishing multiple objectives by a number of agents represented as dynamic systems is considered. Each agent is assumed to have a goal which is to accomplish one or more objectives where each objective is mathematically formulated using an appropriate objective function. Sufficient conditions for accomplishing objectives are derived using particular convergent approximations of minimum and maximum functions depending on the formulation of the goals and objectives. These approximations are differentiable functions and they monotonically converge to the corresponding minimum or maximum function. Finally, an illustrative pursuit-evasion game example with two evaders and two pursuers is provided.     

On Convergence Results for Weak Efficiency in Vector Optimization Problems with Equilibrium Constraints

In this note, we prove that the convergence results for vector optimization problems with equilibrium constraints presented in Wu and Cheng (J. Optim. Theory Appl. 125, 453–472, 2005) are not correct. Actually, we show that results of this type cannot be established at all. This is due to the possible lack, even under nice assumptions, of lower convergence of the solution map for equilibrium problems, already deeply investigated in Loridan and Morgan (Optimization 20, 819–836, 1989) and Lignola and Morgan (J. Optim. Theory Appl. 93, 575–596, 1997).     

Continuity of the Solution Set to Parametric Weak Vector Equilibrium Problems

In this paper, we obtain some stability results for parametric weak vector equilibrium problems in topological vector spaces. We provide sufficient conditions for the continuity of the solution set mapping in parametric weak monotone vector equilibrium problems. This research was partially supported by the National Natural Science Foundation of China (Grant 10561007) and the Natural Science Foundation of Jiangxi Province, China.     

On Existence and Weak Stability of Matrix Refinable Functions

We consider the existence of distributional (or L ₂ ) solutions of the matrix refinement equation where P is an r×r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P (0) has an eigenvalue of the form 2 ⁿ , . A characterization of the existence of L ₂ -solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L ₂ -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997.     

求实际问题中多元函数最值的几点心得

本文在总结多元函数最值求法的基础上,结合实例,指出求解问题中的几个容易被卡住的地方,并给出解决这些问题的几点心得.     

On the Weak Solution of Moving Boundary Problems

The weak formulation of moving boundary problems with possiblyvanishing specific heat, that is governed by parabolic and/orelliptic differential equations, is developed. The uniquenessof the resulting weak solution is then proved. This approachis used to obtain numerical solutions to some physical examples,which arise in electrochemical machining processes, and in saturated/unsaturatedflow in porous media.     

Infine Functions, Nonsmooth Alternative Theorems and Vector Optimization Problems

In this paper we introduce a new notion of infine nonsmooth functions and give several characterizations of infineness property. We prove alternative theorems with mixed constraints (i.e., inequality and equality constraints) being described by invex-infine nonsmooth functions. We establish a necessary and sufficient condition for a solution of a vector optimization problem involving mixed constraints to be a properly efficient solution.     

Derivatives of Set-Valued Maps and Gap Functions for Vector Equilibrium Problems

This paper deals with the set-valued gap functions for vector equilibrium problems and investigates their differential properties using Hadamard directional differentials. Also, contingent and adjacent derivatives of a class of set-valued maps are characterized. Moreover, some basic properties of Φ-contingent and Φ-adjacent cones are given in the presence of a nonsmooth kernel function.     

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.     

Gap Functions and Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems

This paper deals with generalized vector quasi-equilibrium problems. By virtue of a nonlinear scalarization function, the gap functions for two classes of generalized vector quasi-equilibrium problems are obtained. Then, from an existence theorem for a generalized quasi-equilibrium problem and a minimax inequality, existence theorems for two classes of generalized vector quasi-equilibrium problems are established. This research is partially supported by the Postdoctoral Fellowship Scheme of The Hong Kong Polytechnic University and the National Natural Science Foundation of China.     

On Generalized Vector Equilibrium Problems

In this paper we prove the existence of solutions of the generalized vector equilibrium problem in the setting of Hausdorff topological vector spaces. As applications, we present some relevant particular cases: a generalized vector variational-like inequality in Hausdorff topological vector spaces, and equilibrium problem in the case of pseudomonotone real functions, and a generalized weak Pareto optima problem.     

向量优化问题弱有效解集的非空性和紧性刻画

考虑有限维空间Rn(n＞1)中目标映射是仿凸锥映射的向量优化问题.通过对偶锥的端方向和标量函数的0-强制性给出了弱有效解集非空性和紧性的刻画.     

On Generalized Vector Equilibrium Problems

A new generalized vector equilibrium problem involving set-valued mappings and the proper quasi concavity of set-valued mappings in topological vector spaces are introduced; its existence theorems and the convexity of the solution sets are established.     

On Relations Between Vector Optimization Problems and Vector Variational Inequalities

We obtain equivalences between weak Pareto solutions of vector optimization problems and solutions of vector variational inequalities involving generalized directional derivatives.