Generalized Semi-Infinite Programming: Optimality Conditions Involving Reverse Convex Problems

This article deals with a generalized semi-infinite programming problem (S). Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. In particular, a necessary and sufficient optimality condition reduces the problem (S) to a min-max problem constrained with compact convex linked constraints.     

An Efficient Algorithm for Min-Max Convex Semi-Infinite Programming Problems

In this article, we consider the convex min-max problem with infinite constraints. We propose an exchange method to solve the problem by using efficient inactive constraint dropping rules. There is no need to solve the maximization problem over the metric space, as the algorithm has merely to find some points in the metric space such that a certain criterion is satisfied at each iteration. Under some mild assumptions, the proposed algorithm is shown to terminate in a finite number of iterations and to provide an approximate solution to the original problem. Preliminary numerical results with the algorithm are promising. To our knowledge, this article is the first one conceived to apply explicit exchange methods for solving nonlinear semi-infinite convex min-max problems.     

A note on the compactness of the index set in convex optimization. Application to metric regularity

We consider the linear quadratic optimal control problem in the infinite time horizon case for a class of discrete-time systems controlled by a continuous inputs. We show that, under certain hypothesis, the Hilbert uniqueness method can be used to determine the optimal control.     

On Perturbation Stability of Metric Regularity

We introduce a measure of closeness for set-valued mappings from a metric space to a Banach space which allows to quantatively estimate the effect of perturbation of a set-valued mapping on its metric regularity and covering characteristics.     

On Optimality Conditions for Generalized Semi-Infinite Programming Problems

Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary and sufficient conditions for local minimizers of order one without the assumption of local reduction. The necessary conditions are derived along the same lines as the first-order necessary conditions for GSIP in a recent paper of Jongen, Rückmann, and Stein (Ref. 1) by assuming the so-called extended Mangasarian–Fromovitz constraint qualification. Using the ideas of a recent paper of Rückmann and Shapiro, we give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of the Mangasarian–Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem.     

广义半无限规划新的最优性条件

本文讨论了一类指标集依赖于决策变量的广义半无限规划(GSMMP).首先通过刻画目标函数的Clarke导数和Clarke次微分,建立其一阶最优性条件.其次,通过对下层问题Q(x)进行扰动分析,我们得到Q(x)的一个精确罚表示.由此,利用一组精确罚函数将(GSMMP)转化为经典的半无限极大极小规划,从而可利用已有的经典半无限规划的算法来对(GSMMP)进行求解.     

Regularity Conditions in Vector Optimization

By exploiting very recent results concerning linear separation between a set and a convex cone, the image space approach allows us to achieve new regularity conditions for vector minimum problems. In particular, we obtain very general Mangasarian–Fromovitz type conditions.     

Banach空间集值映射的度量正则性与变分方程的Lipschitz稳定性

综述了集值映射的某些概念,例如度量正则性、伪Lipschitz性质(Aubin性质)、度量次正则性和Calm性质和这些概念的相互关系以及某些判据.也给出了他们在变分方程解的鲁棒Lipschitz稳定性、约束优化问题的最优性条件、集合族的线性正则性质和广义方程迭代过程的收敛性.     

First Order Optimality Conditions for Generalized Semi-Infinite Programming Problems

We study first-order optimality conditions for the class of generalized semi-infinite programming problems (GSIPs). We extend various well-known constraint qualifications for finite programming problems to GSIPs and analyze the extent to which a corresponding Karush-Kuhn-Tucker (KKT) condition depends on these extensions. It is shown that in general the KKT condition for GSIPs takes a weaker form unless a certain constraint qualification is satisfied. In the completely convex case where the objective of the lower-level problem is concave and the constraint functions are quasiconvex, we show that the KKT condition takes a sharper form. The authors thank the anonymous referees for careful reading of the paper and helpful suggestions. The research of the first author was partially supported by NSERC.     

Regularity Conditions for the Linear Separation of Sets

In this paper, we study the linear separation between a set and a convex cone. We introduce the concepts of regularity and total regularity of the separation with respect to a face of the cone and we give theorems characterizing them.     

Second-Order Optimality Conditions in Generalized Semi-Infinite Programming

This paper deals with generalized semi-infinite optimization problems where the (infinite) index set of inequality constraints depends on the state variables and all involved functions are twice continuously differentiable. Necessary and sufficient second-order optimality conditions for such problems are derived under assumptions which imply that the corresponding optimal value function is second-order (parabolically) directionally differentiable and second-order epiregular at the considered point. These sufficient conditions are, in particular, equivalent to the second-order growth condition.     

凸度量空间非自混合映射的共同点定理

该文得到了凸度量空间上混合映射的共同点的一些结果.推广了Ciric在文献[3]中的结论.     

Perturbations and Metric Regularity

A point x is an approximate solution of a generalized equation b∈F(x) if the distance from the point b to the set F(x) is small. ‘Metric regularity’ of the set-valued mapping F means that, locally, a constant multiple of this distance bounds the distance from x to an exact solution. The smallest such constant is the ‘modulus of regularity’, and is a measure of the sensitivity or conditioning of the generalized equation. We survey recent approaches to a fundamental characterization of the modulus as the reciprocal of the distance from F to the nearest irregular mapping. We furthermore discuss the sensitivity of the regularity modulus itself, and prove a version of the fundamental characterization for mappings on Riemannian manifolds. Mathematics Subject Classifications 2000 Primary: 49J53; secondary: 90C31.     

Constraint Qualifications Characterizing Lagrangian Duality in Convex Optimization

In convex optimization, a constraint qualification (CQ) is an essential ingredient for the elegant and powerful duality theory. Various constraint qualifications which are sufficient for the Lagrangian duality have been given in the literature. In this paper, we present constraint qualifications which characterize completely the Lagrangian duality.     

Directional End of a Convex Set: Theory and Applications

A point of a convex set belongs to its end in a given direction when this direction is not feasible at that point. This paper analyzes the properties of the directional end of general convex sets and closed convex sets (for which the directional ends are connected by arcs) as well as the relationship between the directional end and certain concepts on the illumination of convex bodies. The paper includes applications of the directional end to the theory of linear systems.     

First-Order Optimality Conditions in Generalized Semi-Infinite Programming

In this paper, we consider a generalized semi-infinite optimization problem where the index set of the corresponding inequality constraints depends on the decision variables and the involved functions are assumed to be continuously differentiable. We derive first-order necessary optimality conditions for such problems by using bounds for the upper and lower directional derivatives of the corresponding optimal value function. In the case where the optimal value function is directly differentiable, we present first-order conditions based on the linearization of the given problem. Finally, we investigate necessary and sufficient first-order conditions by using the calculus of quasidifferentiable functions.     

KKT Optimality Conditions and Nonsmooth Continuous Time Optimization Problems

We introduce a concept of generalized invexity for the nonsmooth continuous time optimization problems, namely, the concept of Karush-Kuhn-Tucker (KKT) invexity. Then, we prove that this notion is necessary and sufficient for global optimality of a KKT point. We also extend the notion of weak-invexity for nonsmooth continuous time optimization problems. Further, we show that weak-invexity is a necessary and sufficient condition for weak duality.     

Geometric Condition Measures and Smoothness Condition Measures for Closed Convex Sets and Linear Regularity of Infinitely Many Closed Convex Sets

In this paper, we study geometric condition measures and smoothness condition measures of closed convex sets, bounded linear regularity, and linear regularity. We show that, under certain conditions, the constant for the linear regularity of infinitely many closed convex sets can be characterized by the geometric condition measure of the intersection or by the smoothness condition measure of the intersection. We study also the bounded linear regularity and present some interesting properties of the general linear regularity problem.The author is grateful to the referees for valuable and constructive suggestions. In particular, she thanks a referee for drawing her attention to Corollary 5.14 of Ref. 3, which inspired her to derive Theorem 4.2 and Corollary 4.2 in the revision of this paper.     

KKT conditions for weak compact convex sets,theorems of the alternative,and optimality conditions

We present several equivalent conditions for the Karush–Kuhn–Tucker conditions for weak^? compact convex sets. Using them, we extend several existing theorems of the alternative in terms of weak^? compact convex sets. Such extensions allow us to express the KKT conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with inequality and equality constraints. Furthermore, several new equivalent optimality conditions for optimization problems with inequality constraints are obtained.     

Lipschitz behavior of convex semi-infinite optimization problems: a variational approach

In this paper we make use of subdifferential calculus and other variational techniques, traced out from [Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 3(333), 103–162; Engligh translation Math. Surveys 55, 501–558 (2000); Ioffe, A.D.: On rubustness of the regularity property of maps. Control cybernet 32, 543–554 (2003)], to derive different expressions for the Lipschitz modulus of the optimal set mapping of canonically perturbed convex semi-infinite optimization problems. In order to apply this background for obtaining the modulus of metric regularity of the associated inverse multifunction, we have to analyze the stable behavior of this inverse mapping. In our semi-infinite framework this analysis entails some specific technical difficulties. We also provide a new expression of a global variational nature for the referred regularity modulus.