一类广义半无限规划问题的一阶最优性条件

本文利用一个精确增广Lagrange函数研究了一类广义半无限极小极大规划问题。在一定的条件下将其转化为标准的半无限极小极大规划问题。研究了这两类问题的最优解和最优值之间的关系,利用这种关系和标准半无限极小极大规划问题的一阶最优性条件给出了这类广义半无限极小极大规划问题的一个新的一阶最优性条件。     

指数型增广拉格朗日函数在广义半无限规划中的应用

本文利用指数型增广拉格朗日函数将一类广义半无限极大极小问题在一定条件下转化为标准的半无限极大极小问题,使它们具有相同的局部与全局最优解.我们给出了两个转化条件:一个是充分与必要条件,另一个是在实际中易于验证的充分条件.通过这种转化,我们给出了广义半无限极大极小问题的一个新的一阶最优性条件.     

半局部凸多目标半无限规划的最优性

研究半局部凸函数在多目标半无限规划下的最优性.利用半局部凸函数,讨论了在多目标半无限规划下的择一定理,最优性条件.使半局部凸函数运用的范围更加广泛.     

一类半无限规划的最优性条件

给出一类广义ηs-凸函数的定义,在其基础上研究并确定广义ηs-凸规划的最优性条件．     

半无限规划的一阶最优性条件和牛顿型算法

在Fischer-Burmeister非线性互补函数的基础上,得到了半无限规划问题的一个新的一阶必要条件,并将半无限规划问题转化成一个光滑的无约束优化问题,给出了适合该问题的一个Damp-Newton算法,数值例子表明:算法结构简单,数值计算有效.     

非紧致集上的最优值函数与广义半无限极大极小规划

研究非紧致集上的最优值函数, 给出了它的方向导数与次微分的结构表示式, 利用它们建立了广义半无限极大极小规划与其一阶最优性条件.     

一类多目标半无限分式规划的最优性条件

<正>0引言分式规划作为最优化的一个分支,近年来,获得了很大的发展,如,文[4]利用(F,α,ρ,d)-凸函数,文[5]利用半局部预不变凸函数等分别讨论了相应的分式规划问题等,这些成果极大地推动了分式规划的发展.     

广义多目标minmax问题的最优性条件和极大熵方法

本文讨论了广义多目标minmax问题的最优性条件。利用极大熵逼近函数，研究了广义多目标minmax；问题的逼近问题，在较弱的条件下，证明了由极大熵逼近函数导出的多目标逼近问题的临界点的任一极限点均为原广义多目标minmax问题的临界点。     

一类非光滑多目标半无限规划的最优性条件

利用K-方向导数,给出了一类存在性更为广泛的广义凸函数.即广义一致K-（F,α,ρ,d）-I型凸函数,进而讨论了涉及这些新广义凸性的一类多目标半无限规划的最优性条件。     

对称弧式连通凸多目标半无限规划的最优性

以弧式连通函数和对称梯度为基础,研究新函数在多目标半无限规划下的最优性理论.定义了一类新的弧式连通函数,对称弧式连通函数、对称拟弧式连通函数、对称弱拟弧式连通函数、对称伪弧式连通函数、对称严格伪弧式连通函数,讨论了这些函数在多目标半无限规划下的最优性.给出更加广义的弧式连通函数,将它们运用到多目标半无限规划.     

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.     

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.     

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.     

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.     

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.     

First-Order Algorithms for Generalized Semi-Infinite Min-Max Problems

We present a first-order algorithm for solving semi-infinite generalized min-max problems which consist of minimizing a function f⁰(x) = F(¹(x), .... , ^m (x)), where F is a smooth function and each ⁱ is the maximum of an infinite number of smooth functions.In Section 3.3 of [17] Polak finds a methodology for solving infinite dimensional problems by expanding them into an infinite sequence of consistent finite dimensional approximating problems, and then using a master algorithm that selects an appropriate subsequence of these problems and applies a number of iterations of a finite dimensional optimization algorithm to each of these problems, sequentially. Our algorithm was constructed within this framework; it calls an algorithm by Kiwiel as a subroutine. The number of iterations of the Kiwiel algorithm to be applied to the approximating problems is determined by a test that ensures that the overall scheme retains the rate of convergence of the Kiwiel algorithm.Under reasonable assumptions we show that all the accumulation points of sequences constructed by our algorithm are stationary, and, under an additional strong convexity assumption, that the Kiwiel algorithm converges at least linearly, and that our algorithm also converges at least linearly, with the same rate constant bounds as Kiwiel's.     

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.     

非光滑半定规划的一阶最优性条件

首次考虑了非光滑半定规化问题．运用与非线性规划类似的技巧,把现存的理论扩展到约束是结构稀疏矩阵的情况,给出了其一阶最优性条件。考虑了严格互补条件不成立的情形．在约束矩阵为对角阵条件下,所用的正则条件与传统非线性优化意义下的是一致的．     

C1,1半定规划的二阶最优性条件

研究了C1,1函数类的半定规划问题,其中的目标函数是连续可微的,其梯度是拟可微的.利用拟可微分析分别给出了二阶必要条件和二阶充分条件.