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.     

On Basic Constraint Qualifications for Infinite System of Convex Inequalities in Banach Spaces

The BCQ and the Abadie CQ for infinite systems of convex inequalities in Banach spaces are characterized in terms of the upper semi-continuity of the convex cones generated by the subdifferentials of active convex functions. Some relationships with other constraint qualifications such as the CPLV and the Slate condition are also studied. Applications in best approximation theory are provided.     

Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and their Local Preservation Property

This paper studies the system of constraint qualifications tailored for mathematical programs with equilibrium constraints. The main focuses are on the relations among them and their local preservation property. After giving a relaxed version of the constant positive linear dependence constraint qualification for mathematical programs with equilibrium constraints, we establish some new relations among the tailored constraint qualifications. Then, we investigate their local preservation property. Finally, we present several results on the isolatedness of local minimizers. The paper contains some proof techniques that seem to be new in the literature of mathematical programs with equilibrium constraints. The obtained results complement and improve some recent ones in this direction.     

Calmness of Linear Constraint Systems under Structured Perturbations with an Application to the Path-Following Scheme

Set-Valued and Variational Analysis - We are concerned with finite linear constraint systems in a parametric framework where the right-hand side is an affine function of the perturbation parameter....     

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.     

具约束的多值映射的calmness的充分与必要条件

本文在—般Banach空间中应用变分分析的手段和方法研究了具集约束的L-subsmooth多值映射具有calmness的充分与必要条件,并得到了Asplund空间中相应的结果.在此基础上,又给出关于此类集值映射具有强calmness的充分条件、必要条件及其等价刻画.最后我们还给出一个模条件判别法.     

Generalized Constraint Qualifications and Optimality Conditions for Set-Valued Optimization Problems

In this paper we discuss the connections of four generalized constraint qualifications for set-valued vector optimization problems with constraints. Then some K-T type necessary and sufficient optimality conditions are derived, in terms of the contingent epiderivatives.     

一类稀疏约束非线性规划的约束规格

研究一类带有闭凸集约束的稀疏约束非线性规划问题,这类问题在变量选择、模式识别、投资组合等领域具有广泛的应用.首先引进了限制性Slater约束规格的概念,证明了该约束规格强于限制性M-F约束规格,然后在此约束规格成立的条件下,分析了其局部最优解成立的充分和必要条件.最后,对约束集合的两种具体形式,指出限制性Slater约束规格必满足,并给出了一阶必要性条件的具体表达形式.     

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.      

DC Optimization Approach to Metric Regularity of Convex Multifunctions with Applications to Infinite Systems

The paper develops a new approach to the study of metric regularity and related well-posedness properties of convex set-valued mappings between general Banach spaces by reducing them to unconstrained minimization problems with objectives given as the difference of convex (DC) functions. In this way, we establish new formulas for calculating the exact regularity bound of closed and convex multifunctions and apply them to deriving explicit conditions ensuring well-posedness of infinite convex systems described by inequality and equality constraints.     

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.

     

Calmness of the Feasible Set Mapping for Linear Inequality Systems

In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system’s data.     

New Results on Constraint Qualifications for Nonlinear Extremum Problems and Extensions

In this paper, we focus on some new constraint qualifications introduced for nonlinear extremum problems in the recent literature. We show that, if the constraint functions are continuously differentiable, the relaxed Mangasarian–Fromovitz constraint qualification (or, equivalently, the constant rank of the subspace component condition) implies the existence of local error bounds for the system of inequalities and equalities. We further extend the new result to the mathematical programs with equilibrium constraints. In particular, we show that the MPEC relaxed (or enhanced relaxed) constant positive linear dependence condition implies the existence of local error bounds for the mixed complementarity system.     

Theorems of the Alternative for Multivalued Mappings and Applications to Mixed Convex\Concave Systems of Inequalities

We provide a purely algebraic theorem of the alternative (and its topological variant) involving multivalued mappings under relaxed convexity assumptions. Various more or less classical applications are given, specially for nonconvex quadratic systems. In the second part of the paper we introduce an alternative formulation for a mixed convex\concave statement. The theory is applied to systems of mixed convex\concave inequalities.     

A Practical Optimality Condition Without Constraint Qualifications for Nonlinear Programming

A new optimality condition for minimization with general constraints is introduced. Unlike the KKT conditions, the new condition is satisfied by local minimizers of nonlinear programming problems, independently of constraint qualifications. The new condition is strictly stronger than and implies the Fritz–John optimality conditions. Sufficiency for convex programming is proved.     

On Constraint Qualifications and Sensitivity Analysis for General Optimization Problems via Pseudo-Jacobians

A nonsmooth and nonconvex general optimization problem is considered. Using the idea of pseudo-Jacobians, nonsmooth versions of the Robinson and Mangasarian–Fromovitz constraint qualifications are defined and relationships between them and the local error bound property are investigated. A new necessary optimality condition, based on the pseudo-Jacobians, is derived under the local error bound constraint qualification. These results are applied for computing and estimating the Fréchet and limiting subdifferentials of value functions. Moreover, several examples are provided to clarify the obtained results.     

Constraint Qualifications and Stationary Conditions for Mathematical Programming with Non-differentiable Vanishing Constraints

This paper aims at studying a broad class of mathematical programming with non-differentiable vanishing constraints. First, we are interested in some various qualification conditions for the problem. Then, these constraint qualifications are applied to obtain, under different conditions, several stationary conditions of type Karush/Kuhn–Tucker.     

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.     

Complementarity Constraint Qualifications and Simplified B-Stationarity Conditions for Mathematical Programs with Equilibrium Constraints

With the aid of some novel complementarity constraint qualifications, we derive some simplified primal-dual characterizations of a B-stationary point for a mathematical program with complementarity constraints (MPEC). The approach is based on a locally equivalent piecewise formulation of such a program near a feasible point. The simplified results, which rely heavily on a careful dissection and improved understanding of the tangent cone of the feasible region of the program, bypass the combinatorial characterization that is intrinsic to B-stationarity.