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。     

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.     

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.     

半无限极大极小问题的全局收敛方法

利用广义伪方向导数，在较弱的条件下，给出了半无限极大极小问题（P）的全局收敛性理论算法模型；利用离散策略给出了问题（P）全局收敛的可实现算法．数值结果表明本文给出的可实现算法是有效的．     

关于一致半格的注记

首先证明了满足极大条件的无限链一定是一致半格.其次,通过探究一致半格与极大条件之间的关系,给出了一致半格为满足极大条件的无限链的充分必要条件.最后,讨论了一致半格与极小条件以及反一致半格与极大条件之间的内在关系,并且给出了相应的例子.     

一类广义半无限规划问题的光滑牛顿算法

本文在广义半无限规划问题的最优解集X处满足某些条件的前提下将广义半无限规划问题转化成KKT系统,通过扰动的FB函数,将KKT系统转化为一组光滑函数方程,设计了一个光滑牛顿算法,证明了算法的全局收敛性,并且在光滑函数解集处满足局部误差界条件下证明了算法具有超线性收敛速率.     

广义半无限规划问题的一种可行方向算法

基于Zoutendijk可行方向算法,本文提出了一种求解广义半无限规划问题的可行方向算法,在保证算法收敛的情况下,此算法比以往的算法在假设条件的要求上有着一定的优势,且数值试验表明此法是可行的.     

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.     

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.     

凸半无限规划的一个新的割平面算法

基于非线性规划和割平面方法,给出了凸半无限规划问题的一个分析中央割平面算法(ACCPM).该算法不需要在每一次迭代时计算所有的约束数值,而只需要求解一个中央割平面,从而使得问题的求解规模变小,这种算法对于求解可行域结构比较复杂的半无限规划非常有效,最后给出算法的收敛性证明.     

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.     

On the Use of Augmented Lagrangians in the Solution of Generalized Semi-Infinite Min-Max Problems

We present an approach for the solution of a class of generalized semi-infinite optimization problems. Our approach uses augmented Lagrangians to transform generalized semi-infinite min-max problems into ordinary semi-infinite min-max problems, with the same set of local and global solutions as well as the same stationary points. Once the transformation is effected, the generalized semi-infinite min-max problems can be solved using any available semi-infinite optimization algorithm. We illustrate our approach with two numerical examples, one of which deals with structural design subject to reliability constraints.     

Optimal value functions of generalized semi-infinite min-max programming on a noncompact set

In this paper, we study optimal value functions of generalized semi-infinite min-max programming problems on a noncompact set. Directional derivatives and subd-ifferential characterizations of optimal value functions are given. Using these properties, we establish first order optimality conditions for unconstrained generalized semi-infinite programming problems.     

On global quadratic growth condition for min-max optimization problems with quadratic functions

Second-order sufficient condition and quadratic growth condition play important roles both in sensitivity and stability analysis and in numerical analysis for optimization problems. In this article, we concentrate on the global quadratic growth condition and study its relations with global second-order sufficient conditions for min-max optimization problems with quadratic functions. In general, the global second-order sufficient condition implies the global quadratic growth condition. In the case of two quadratic functions involved, we have the equivalence of the two conditions.     

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.     

Second order optimality conditions for generalized semi-infinite programming problems

We consider generalized semi-infinite programming problems. Second order necessary and sufficient conditionsfor local optimality are given. The conditions are derived under assumptions such that the feasible set can be described by means of a finite number of optimal value functions. Since we do not require a strict complementary condition for the local reduction these functions are only of class C¹ A sufficient condition for optimality is proven under much weaker assumptions.     

Optimality conditions for non-smooth semi-infinite programming

This article deals with a class of non-smooth semi-infinite programming (SIP) problems in which the index set of the inequality constraints is an arbitrary set not necessarily finite. We introduce several kinds of constraint qualifications for these non-smooth SIP problems and we study the relationships between them. Finally, necessary and sufficient optimality conditions are investigated.     

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.