The Tangent cones on constraint qualifications in optimization problems

This article proposes a few tangent cones,which are relative to the constraint qualifications of optimization problems.With the upper and lower directional derivatives of an objective function,the characteristics of cones on the constraint qualifications are presented.The interrelations among the constraint qualifications,a few cones involved, and level sets of upper and lower directional derivatives are derived.     

Constraint qualifications for constrained Lipschitz optimization problems and applications to a MPCC

Three constraint qualifications (the weak generalized Robinson constraint qualification, the bounded constraint qualification, and the generalized Abadie constraint qualification), which are weaker than the generalized Robinson constraint qualification (GRCQ) given by Yen (1997) [1], are introduced for constrained Lipschitz optimization problems. Relationships between those constraint qualifications and the calmness of the solution mapping are investigated. It is demonstrated that the weak generalized Robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping, whereas the proposed generalized Abadie constraint qualification, described in terms of graphical derivatives in variational analysis, is weaker than the calmness of the solution mapping. Finally, those constraint qualifications are written for a mathematical program with complementarity constraints (MPCC), and new constraint qualifications ensuring the C-stationary point condition of a MPCC are obtained.     

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.

     

Necessary and Sufficient Constraint Qualification for Surrogate Duality

In mathematical programming, constraint qualifications are essential elements for duality theory. Recently, necessary and sufficient constraint qualifications for Lagrange duality results have been investigated. Also, surrogate duality enables one to replace the problem by a simpler one in which the constraint function is a scalar one. However, as far as we know, a necessary and sufficient constraint qualification for surrogate duality has not been proposed yet. In this paper, we propose necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, which are closely related with ones for Lagrange duality.     

On Several Types of Basic Constraint Qualifications via Coderivatives for Generalized Equations

In this paper, we study several types of basic constraint qualifications in terms of Clarke/Fréchet coderivatives for generalized equations. Several necessary and/or sufficient conditions are given to ensure these constraint qualifications. It is proved that basic constraint qualification and strong basic constraint qualification for convex generalized equations can be obtained by these constraint qualifications, and the existing results on constraint qualifications for the inequality system can be deduced from the given conditions in this paper. The main work of this paper is an extension of the study on constraint qualifications from inequality systems to generalized equations.     

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.     

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.     

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     

Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints

We study second-order optimality conditions for mathematical programs with equilibrium constraints (MPEC). Firstly, we improve some second-order optimality conditions for standard nonlinear programming problems using some newly discovered constraint qualifications in the literature, and apply them to MPEC. Then, we introduce some MPEC variants of these new constraint qualifications, which are all weaker than the MPEC linear independence constraint qualification, and derive several second-order optimality conditions for MPEC under the new MPEC constraint qualifications. Finally, we discuss the isolatedness of local minimizers for MPEC under very weak conditions.     

On constraint qualifications in terms of approximate Jacobians for nonsmooth continuous optimization problems

Constraint qualifications in terms of approximate Jacobians are investigated for a nonsmooth constrained optimization problem, in which the involved functions are continuous but not necessarily locally Lipschitz. New constraint qualifications in terms of approximate Jacobians, weaker than the generalized Robinson constraint qualification (GRCQ) in Jeyakumar and Yen [V. Jeyakumar, N.D. Yen, Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization, SIAM J. Optim. 14 5 (2004) 1106-1127], are introduced and some examples are provided to show the utility of constrained qualifications introduced. Since the calmness condition is regarded as the basic condition for optimality conditions, the relationships between the constraint qualifications proposed and the calmness of solution mapping are also studied.     

Abadie Constraint Qualifications for Convex Constraint Systems and Applications to Calmness Property

In this paper, we mainly study concepts of Abadie constraint qualification and strong Abadie constraint qualification for a convex constraint system defined by a closed convex multifunction and a closed convex subset. These concepts can cover Abadie constraint qualifications for the feasible region of convex optimization problem and the convex multifunction. Several characterizations for these Abadie constraint qualifications are also provided. As applications, we use these Abadie constraint qualifications to characterize calmness properties of the convex constraint system.     

On Error Bounds for Quasinormal Programs

The Mangasarian-Fromovitz constraint qualification is a central concept within the theory of constraint qualifications in nonlinear optimization. Nevertheless there are problems where this condition does not hold though other constraint qualifications can be fulfilled. One of such constraint qualifications is the so-called quasinormality by Hestenes. The well known error bound property (R-regularity) can also play the role of a general constraint qualification providing the existence of Lagrange multipliers. In this note we investigate the relation between some constraint qualifications and prove that quasinormality implies the error bound property, while the reciprocal is not true.     

BCQ and Strong BCQ for Nonconvex Generalized Equations with Applications to Metric Subregularity

In this paper, based on basic constraint qualification (BCQ) and strong BCQ for convex generalized equation, we are inspired to further discuss constraint qualifications of BCQ and strong BCQ for nonconvex generalized equation and then establish their various characterizations. As applications, we use these constraint qualifications to study metric subregularity of nonconvex generalized equation and provide necessary and/or sufficient conditions in terms of constraint qualifications considered herein to ensure nonconvex generalized equation having metric subregularity.     

Existence and Boundedness of the Kuhn-Tucker Multipliers in Nonsmooth Multiobjective Optimization

Using the idea of upper convexificators, we propose constraint qualifications and study existence and boundedness of the Kuhn-Tucker multipliers for a nonsmooth multiobjective optimization problem with inequality constraints and an arbitrary set constraint. We show that, at locally weak efficient solutions where the objective and constraint functions are locally Lipschitz, the constraint qualifications are necessary and sufficient conditions for the Kuhn-Tucker multiplier sets to be nonempty and bounded under certain semiregularity assumptions on the upper convexificators of the functions.     

On relaxing the Mangasarian–Fromovitz constraint qualification

For the classical nonlinear program, two new relaxations of the Mangasarian–Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an application, we make use of this new constraint qualification in the local analysis of the solution map to a parameterized equilibrium problem, modeled by a generalized equation.     

Necessary Optimality Conditions in Terms of Convexificators in Lipschitz Optimization

This study is devoted to constraint qualifications and Kuhn-Tucker type necessary optimality conditions for nonsmooth optimization problems involving locally Lipschitz functions. The main tool of the study is the concept of convexificators. First, the case of a minimization problem in the presence of an arbitrary set constraint is considered by using the contingent cone and the adjacent cone to the constraint set. Then, in the case of a minimization problem with inequality constraints, Abadie type constraint qualifications and several other qualifications are proposed; Kuhn-Tucker type necessary optimality conditions are derived under the qualifications.Communicated by S. SchaibleThe authors thank the referees for bringing to their attention some papers closely related to this study and for helpful comments and constructive suggestions that have greatly improved the original version of the paper. Further, they are indebted to Professors H. W. Sun and F. Y. Lu, who suggested an example for this paper. The first author thanks S. Schaible for encouragement during this research.     

Notes on Some Constraint Qualifications for Mathematical Programs with Equilibrium Constraints

We study the constraint qualifications for mathematical programs with equilibrium constraints (MPEC). Firstly, we investigate the weakest constraint qualifications for the Bouligand and Mordukhovich stationarities for MPEC. Then, we show that the MPEC relaxed constant positive linear dependence condition can ensure any locally optimal solution to be Mordukhovich stationary. Finally, we give the relations among the existing MPEC constraint qualifications.     

Constraint Qualifications for Nonsmooth Mathematical Programs with Equilibrium Constraints

We study nonsmooth mathematical programs with equilibrium constraints. First we consider a general disjunctive program which embeds a large class of problems with equilibrium constraints. Then, we establish several constraint qualifications for these optimization problems. In particular, we generalize the Abadie and Guignard-type constraint qualifications. Subsequently, we specialize these results to mathematical program with equilibrium constraints. In our investigation, we show that a local minimum results in a so-called M-stationary point under a very weak constraint qualification.      

Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems

In this paper, we present several constraint qualifications, and we show that these conditions guarantee the nonvacuity and the boundedness of the Lagrange multiplier sets for general nondifferentiable programming problems. The relationships with various constraint qualifications are investigated.The author gratefully acknowledges the comments made by the two referees.     

A Fritz John Approach to First Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints

Mathematical programs with equilibrium constraints (MPECs) are nonlinear programs which do not satisfy any of the common constraint qualifications. In order to obtain first order optimality conditions, constraint qualifications tailored to MPECs have been developed and researched in the past. This has been done by falling back on technical proofs or results from nonsmooth analysis. In this article, we use a completely different approach and show how the standard Fritz John conditions may be used in order to obtain short and elementary proofs for the most important optimality conditions for MPECs. As a by-product, we obtain a new stationarity concept.