均衡约束数学规划的约束规格和最优性条件综述

约束规格在约束优化问题的最优性条件中起着重要的作用,介绍了近几年国际上关于均衡约束数学规划(简记为MPEC)的约束规格以及最优性条件的研究成果, 包括以下主要内容: (1) MPEC常用的约束规格(如线性无关约束规格 (MPEC-LICQ)、Mangasarian-Fromovitz约束规格 (MPEC-MFCQ)等)和新的约束规格(如恒秩约束规格、常数正线性相关约束规格等), 以及它们之间的关系; (2) MPEC常用的稳定点; (3) MPEC的最优性条件. 最后还对MPEC的约束规格和最优性条件的研究前景进行了探讨.     

均衡约束数学规划问题的一种新的约束规格

本文研究了均衡约束数学规划(MPEC)问题.利用其弱稳定点,获得了一种新的约束规格–MPEC的伪正规约束规格.用一种简单的方式,证明了该约束规格是介于MPEC-MFCQ(即MPEC,Mangasarian-Fromowitz约束规格)与MPEC-ACQ(即MPEC,Abadie约束规格)之间的约束规格,因此该约束规格也可以导出MPEC问题的M-稳定点.最后通过两个例子,说明了该约束规格与MPEC-MFCQ以及与MPEC-ACQ之间是严格的强弱关系.     

关于双层规划约束规格的一个注记

本文表明了非线性规划中常见的约束规格对一般双层规划不成立,并对双层规划可以满足的较弱的约束规格“部分平静”,给出了使其成立的充分条件．     

向量映射的鞍点和Lagrange对偶问题

本文研究拓扑向量空间广义锥-次类凸映射向量优化问题的鞍点最优性条件和Lagrange对偶问题,建立向量优化问题的Fritz John鞍点和Kuhn-Tucker鞍点的最优性条件及其与向量优化问题的有效解和弱有效解之间的联系。通过对偶问题和向量优化问题的标量化刻画各解之间的关系,给出目标映射是广义锥-次类凸的向量优化问题在其约束映射满足广义Slater约束规格的条件下的对偶定理。     

关于求解带线性互补约束的数学规划问题正则方法的一个注记

本文主要研究在某些较弱条件下求解带线性互补约束的数学规划问题(MPLCC)正则方法的收敛性.若衡约束规划线性独立约束规范条件(MPEC-LICQ)在由正则方法产生的点列的聚点处成立,且迭代点列满足二阶必要条件,同时,若比在文[7]中渐近弱非退化条件Ⅰ更弱的渐近弱非退化条件Ⅱ在聚点处也成立,那么所有这些聚点都是B-稳定点.此外,在弱MPEC-LICQ,二阶必要条件及渐近弱退化条件Ⅱ下,我们仍能证明通过正则方法所得的聚点都是B-稳定点.     

较多约束多目标规划的最优性条件

本文研究较多约束多目标规划的最优性条件.借助于所给问题的较多约束集结构表示,定义了较多约束规划问题的较多约束Pareto有效解和较多约束Pareto弱有效解,给出较多约束Pareto有效解和较多约束Pareto弱有效解要满足的Fritz John条件和Kuhn-Tucker条件,最后给出在凸性条件下它的一些最优性充分条件.     

混料试验中的确界约束与多余约束

在混料问题中,经常出现附加的混料约束.对于单分量上、下界约束系,我们证明:只存在两种非确界约束,或者都是单侧的,或者只有一个分量存在双侧非确界约束,不可能有两个分量存在异侧非确界约束.对于分量线性组合上、下界约束系和复合约束系,我们分别给出了关于多余约束的判别方法,并用了一些数值例子说明这些判别方法.     

集值优化问题的Benson真有效解的广义导数型最优性条件

引进了集值映射关于锥的Clarke切导数, Adjacent切导数与Contingent切导数概念;应用它们导出了具Slater约束规格的集值优化问题的Benson真有效解的广义导数型最优性条件．     

约束优化问题中常用的约束规范及其相互关系

详细分析了约束优化问题中几种常见的约束规范,如L ICQ,SM FCQ,M FCQ,CRCQ,CPLD以及伪正规,拟正规和拟正则约束规范.针对等式和不等式约束问题讨论了它们与拉格朗日乘子的存在性及其性质之间的关系,给出了各种约束规范之间的关系图.特别通过反例,说明了WM FCQ在含等式约束的问题中不是一种约束规范.     

一个求解消失约束数学规划的部分光滑正则化方法（英文）

本文研究带有消失约束的数学规划问题.针对这一问题,我们提出了一种基于伪Huber函数的光滑正则化方法,该方法只对部分消失约束进行光滑化.对于新的光滑问题,我们证明Mangasarian-Fromovitz约束规格在某些情况下是成立的.我们也分析该方法的收敛性质,即,一个光滑正则化问题稳定点序列的聚点是原问题的T-稳定点,并给出光滑正则化问题稳定点序列的聚点是原问题的M-稳定点或S-稳定点的一些充分条件.最后初步的数值结果表明该方法是可行的.     

Refinements of necessary conditions for optimality in nonlinear programming

In this paper, a new set of necessary conditions for optimality is introduced with reference to the differentiable nonlinear programming problem. It is shown that these necessary conditions are sharper than the usual Fritz John ones. A constraint qualification relevant to the new necessary conditions is defined and extensions to the locally Lipschitz case are presented.     

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.     

集值映射多目标规划的K-T最优性条件

讨论集值映射多目标规划(VP)的最优性条件问题.首先,在没有锥凹的假设下,利用集值映射的相依导数,得到了(VP)的锥--超有效解要满足的必要条件和充分条件.其次,在锥凹假设和比推广了的Slater规格更弱的条件下,给出了(VP)关于锥--超有效解的K--T型最优性必要条件和充分条件.     

Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems

This paper is devoted to the study of nonsmooth generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be locally Lipschitz. We introduce a constraint qualification which is based on the Mordukhovich subdifferential. Then, we derive a Fritz–John type necessary optimality condition. Finally, interrelations between the new and the existing constraint qualifications such as the Mangasarian–Fromovitz, linear independent, and the Slater are investigated.     

Nonsmooth multiobjective programming and constraint qualifications

We consider a nonsmooth multiobjective programming problem with inequality and set constraints. By using the notion of convexificator, we extend the Abadie constraint qualification, and derive the strong Kuhn-Tucker necessary optimality conditions. Some other constraint qualifications have been generalized and their interrelations are investigated.     

Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints

Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper, we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ, discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ.Communicated by Z. Q. Luo     

Duality Theorems for Separable Convex Programming Without Qualifications

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.     

New Constraint Qualifications and Optimality Conditions for Second Order Cone Programs

We introduce three new constraint qualifications for nonlinear second order cone programming problems that we call constant rank constraint qualification, relaxed constant rank constraint qualification and constant rank of the subspace component condition. Our development is inspired by the corresponding constraint qualifications for nonlinear programming problems. We provide proofs and examples that show the relations of the three new constraint qualifications with other known constraint qualifications. In particular, the new constraint qualifications neither imply nor are implied by Robinson’s constraint qualification, but they are stronger than Abadie’s constraint qualification. First order necessary optimality conditions are shown to hold under the three new constraint qualifications, whereas the second order necessary conditions hold for two of them, the constant rank constraint qualification and the relaxed constant rank constraint qualification.

     

一类非光滑广义分式规划的Kuhn-Tucker型最优性必要条件

考虑一类非线性不等式约束的非光滑minimax分式规划问题;目标函数中的分子是可微函数与凸函数之和形式而分母是可微函数与凸函数之差形式,且约束函数是可微的.在Arrow- Hurwicz-Uzawa约束品性下,给出了这类规划的最优解的Kuhn-Tucker型必要条件.所得结果改进和推广了已有文献中的相应结果.     

Trust-region problems with linear inequality constraints: exact SDP relaxation,global optimality and robust optimization

The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.