On Constraint Qualifications in Nonsmooth Optimization

We introduce extensions of the Mangasarian-Fromovitz and Abadie constraint qualifications to nonsmooth optimization problems with feasibility given by means of lower-level sets. We do not assume directional differentiability, but only upper semicontinuity of the defining functions. By deriving and reviewing primal first-order optimality conditions for nonsmooth problems, we motivate the formulations of the constraint qualifications. Then, we study their interrelation, and we show how they are related to the Slater condition for nonsmooth convex problems, to nonsmooth reverse-convex problems, to the stability of parametric feasible set mappings, and to alternative theorems for the derivation of dual first-order optimality conditions.In the literature on general semi-infinite programming problems, a number of formally different extensions of the Mangasarian-Fromovitz constraint qualification have been introduced recently under different structural assumptions. We show that all these extensions are unified by the constraint qualification presented here.     

Nondifferentiable optimization in banach space

Necessary optimality conditions for nondifferentiable optimization problems in Banach spaces with its objective function and constraint function valued in a partial ordered vector space are given. The main results are obtained with the help of several Farkas type alternative theorems, one of them is proven in the paper for the first time only under a strict separation axiom of convex sets.     

A closedness condition and its applications to DC programs with convex constraints

This paper concerns a closedness condition called (CC) involving a convex function and a convex constrained system. This type of condition has played an important role in the study of convex optimization problems. Our aim is to establish several characterizations of this condition and to apply them to study problems of minimizing a DC function under a cone-convex constraint and a set constraint. First, we establish several so-called ‘Toland–Fenchel–Lagrange’ duality theorems. As consequences, various versions of generalized Farkas lemmas in dual forms for systems involving convex and DC functions are derived. Then, we establish optimality conditions for DC problem under convex constraints. Optimality conditions for convex problems and problems of maximizing a convex function under convex constraints are given as well. Most of the results are established under the (CC) condition. This article serves as a link between several corresponding known ones published recently for DC programs and for convex programs.     

Infinite Alternative Theorems and Nonsmooth Constraint Qualification Conditions

In this paper, some infinite alternative theorems in locally convex vector spaces are proved and some applications of these theorems in optimization theory are addressed. Extensions of constraint qualification conditions, in the absence of differentiability using Clarke??s generalized gradient and Mordukhovich??s subdifferential, in Banach and Asplund spaces are introduced; and relationships between them are established. Some of these relations, are proved using the provided alternative theorems.     

关于一类向量极值问题弱有效解的充分必要条件

本文研究带集合约束的向量极值问题。运用局部凸Hausdorff拓扑向量空间中广义次似凸映射的择一定理和其他一些结论，得到了关于集合约束向量极值问题弱有效解的几个充分必要条件．     

微积分中几个定理的另类处理

在本文中,我们转换视角考察微积分中几个重要定理之间的关系,给出这几个定理的非传统的另类证明和新的蕴涵关系以及彼此等价的结论。     

General existence theorems, alternative theorems and applications to minimax problems

We establish general theorems on maximal elements, coincidence points and nonempty intersections for set-valued mappings on GFC-spaces and show their equivalence. Applying them we derive equivalent forms of alternative theorems. As applications, we develop in detail general types of minimax theorems. The results obtained improve or include as special cases several recent ones in the literature.     

Nonsmooth multiobjective programming: strong Kuhn–Tucker conditions

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond–Weir type dual problems are formulated. We also establish the weak and strong duality theorems.     

On alternative theorems and necessary conditions for efficiency

In this article, we establish theorems of the alternative for a system described by inequalities, equalities and a set inclusion, which are generalizations of Tucker's classical theorem of the alternative, and develop Kuhn–Tucker necessary conditions for efficiency to mathematical programs in normed linear spaces involving inequality, equality and set constraints with positive Lagrange multipliers of all the components of objective functions.     

Penalty Function Approach to Linear Trilevel Programming

In this paper, we study a trilevel decision-making situation in which decisionmaker 1 selects an action, within a specified constraint set, then decisionmaker 2 selects an action within a constraint set determined by the action of decisionmaker 2, and finally decisionmaker 3 selects an action within a constraint set determined by the actions of decisionmakers 1 and 2. Each decisionmaker has an objective function to be optimized within the imposed constraint set. Bard (Ref. 1) presents a five-step algorithm for solving this problem. This paper presents an alternative approach to the key first step of that algorithm, which has some qualitative advantages over it.     

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems

In this paper, we first establish the existence theorems of generalized vector quasi-equilibrium problems. From these results, we establish the existence theorems of common fixed point theorems for two multivalued maps and mathematical programs with an equilibrium constraint as applications.     

Existence Theorems of Quasivariational Inclusion Problems with Applications to Bilevel Problems and Mathematical Programs with Equilibrium Constraint

In this paper, we establish existence theorems of quasivariational inclusion problems; from them, we establish existence theorems of mathematical programs with quasivariational inclusion constraint, bilevel problems, mathematical programs with equilibrium constraint and semi-infinite problems. This research was supported by the National Science Council of the Republic of China. The authors express their gradtitude to the referees for valuable suggestions.     

Nonlinear separation approach to inverse variational inequalities

In this paper, we employ the image space analysis to investigate an inverse variational inequality (for short, IVI) with a cone constraint. By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are first introduced. Then, by these nonlinear separation functions, theorems of the weak and strong alternative and some optimality conditions for IVI with a cone constraint are derived without any convexity. In particular, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, two gap functions and an error bound for IVI with a cone constraint are obtained.     

Pairs of monotone operators

This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent.

     

Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps

This paper extends the concept of cone subconvexlikeness of single-valued maps to set-valued maps and presents several equivalent characterizations and an alternative theorem for cone-subconvexlike set-valued maps. The concept and results are then applied to study the Benson proper efficiency for a vector optimization problem with set-valued maps in topological vector spaces. Two scalarization theorems and two Lagrange multiplier theorems are established. After introducing the new concept of proper saddle point for an appropriate set-valued Lagrange map, we use it to characterize the Benson proper efficiency. Lagrange duality theorems are also obtained     

On the dependence of the solutions and optimal solutions of control problems on the control constraint set

In this paper we examine the dependence of the solutions and optimal solutions of a class of linear, infinite dimensional control systems on the control constraint set. This is done using the weak and the Kuratowski-Mosco convergence of sets. First we establish some general facts about weakly convergent multifunctions. Then we prove some convergence theorems for the trajectories of certain control systems. We also derive a general relaxation theorem. Subsequently we pass to optimal control problems and prove various convergence results. We conclude with an example from parabolic control systems.Research supported by N. S. F. Grant DMS-8802688     

Exceptional family and solution existence of variational inequality problems with set-valued mappings

In this paper, we introduce a new exceptional family for a variational inequality with a set-valued mapping over a general unbounded closed convex set in a Hilbert space. By means of the exceptional family and topological degree theory of set-valued mappings, an alternative theorem and some solution existence theorems are obtained.     

The relationship between theorems of the alternative,least norm problems,steepest descent directions,and degeneracy: A review

Recently, it has been observed that several nondifferentiable minimization problems share the property that the question of whether a given point is optimal can be answered by solving a certain bounded least squares problem. If the resulting residual vector,r, vanishes then the current point is optimal. Otherwise,r is a descent direction. In fact, as we shall see,r points at the steepest descent direction. On the other hand, it is customary to characterize the optimality conditions (and the steepest descent vector) of a convex nondifferentiable function via its subdifferential. Also, it is well known that optimality conditions are usually related to theorems of the alternative. One aim of our survey is to clarify the relations between these subjects. Another aim is to introduce a new type of theorems of the alternative. The new theorems characterize the optimality conditions of discretel ₁ approximation problems and multifacility location problems, and provide a simple way to obtain the subdifferential and the steepest descent direction in such problems. A further objective of our review is to demonstrate that the ability to compute the steepest descent direction at degenerate dead points opens a new way for handling degeneracy in active set methods.     

The recognition and the constitution of the theorems of closure

This papers analyzes how several geometric theorems, that were considered to be disconnected from each other in the beginning of the nineteenth century, have been progressively recognized as elements of a bigger whole called “the theorems of closure.” In particular, we show that the constitution of this set of theorems was grounded on the use of encompassing words, as well as observations of analogies, and searches for unifying points of view. In the concluding remarks, we discuss the relevancy of the notion of “family resemblance” to describe the categorization process of the theorems of closure during the nineteenth century.