Relation between the constant rank and the relaxed constant rank constraint qualifications

This article discusses a relation between the constant rank constraint qualification (CRCQ) and the recently proposed relaxed constant rank constraint qualification (RCRCQ). We show that a parametric constraint system satisfying the RCRCQ is locally diffeomorphic to a system satisfying the CRCQ. We use this result to extend some existing results for the CRCQ to the RCRCQ, establish a relation between the RCRCQ and the Mangasarian–Fromovitz constraint qualification, and obtain a weakened version of the Aubin property under the RCRCQ.     

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.

     

On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification

The constant positive linear dependence (CPLD) condition for feasible points of nonlinear programming problems was introduced by Qi and Wei (Ref. 1) and used in the analysis of SQP methods. In that paper, the authors conjectured that the CPLD could be a constraint qualification. This conjecture is proven in the present paper. Moreover, it is shown that the CPLD condition implies the quasinormality constraint qualification, but that the reciprocal is not true. Relations with other constraint qualifications are given.This research has been supported by PRONEX-Optimization Grant 76.79.1008-00, by FAPESP Grants 01-04597-4 and 02-00832-1, and by CNPq. The authors are indebted to two anonymous referees for useful comments and to Prof. Liqun Qi for encouragement.     

A MULTIDIMENSIONAL FILTER SQP ALGORITHM FOR NONLINEAR PROGRAMMING

We propose a multidimensional filter SQP algorithm. The multidimensional filter technique proposed by Gould et al. [SIAM J. Optim., 2005] is extended to solve constrained optimization problems. In our proposed algorithm, the constraints are partitioned into several parts, and the entry of our filter consists of these different parts. Not only the criteria for accepting a trial step would be relaxed, but the individual behavior of each part of constraints is considered. One feature is that the undesirable link between the objective function and the constraint violation in the filter acceptance criteria disappears. The other is that feasibility restoration phases are unnecessary because a consistent quadratic programming subproblem is used. We prove that our algorithm is globally convergent to KKT points under the constant positive generators (CPG) condition which is weaker than the well-known Mangasarian-Fromovitz constraint qualification (MFCQ) and the constant positive linear dependence (CPLD). Numerical results are presented to show the efficiency of the algorithm.     

A Relaxed Constant Positive Linear Dependence Constraint Qualification for Mathematical Programs with Equilibrium Constraints

We introduce a relaxed version of the constant positive linear dependence constraint qualification for mathematical programs with equilibrium constraints (MPEC). This condition is weaker but easier to check than the MPEC constant positive linear dependence constraint qualification, and stronger than the MPEC Abadie constraint qualification (thus, it is an MPEC constraint qualification for M-stationarity). Neither the new constraint qualification implies the MPEC generalized quasinormality, nor the MPEC generalized quasinormality implies the new constraint qualification. The new one ensures the validity of the local MPEC error bound under certain additional assumptions. We also have improved some recent results on the existence of a local error bound in the standard nonlinear program.     

Optimality Conditions for Disjunctive Programs with Application to Mathematical Programs with Equilibrium Constraints

We consider optimization problems with a disjunctive structure of the feasible set. Using Guignard-type constraint qualifications for these optimization problems and exploiting some results for the limiting normal cone by Mordukhovich, we derive different optimality conditions. Furthermore, we specialize these results to mathematical programs with equilibrium constraints. In particular, we show that a new constraint qualification, weaker than any other constraint qualification used in the literature, is enough in order to show that a local minimum results in a so-called M-stationary point. Additional assumptions are also discussed which guarantee that such an M-stationary point is in fact a strongly stationary point.      

Some kind of Pareto stationarity for multiobjective problems with equilibrium constraints

ABSTRACT

In this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that the generalized Guignard constraint qualification is the weakest constraint qualification for the strong Pareto Kuhn-Tucker optimality. Furthermore, under certain convexity or generalized convexity assumptions, we show that the strong Pareto Kuhn-Tucker optimality conditions are also sufficient for several popular locally Pareto-type optimality conditions for MOPEC.     

Newton’s method for computing a normalized equilibrium in the generalized Nash game through fixed point formulation

We consider the generalized Nash equilibrium problem (GNEP), where not only the players’ cost functions but also their strategy spaces depend on the rivals’ decision variables. Existence results for GNEPs are typically shown by using a fixed point argument for a certain set-valued function. Here we use a regularization of this set-valued function in order to obtain a single-valued function that is easier to deal with from a numerical point of view. We show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria. This fixed point formulation is then used to develop a nonsmooth Newton method for computing a normalized equilibrium. The method uses a so-called computable generalized Jacobian that is much easier to compute than Clarke generalized Jacobian or B-subdifferential. We establish local superlinear/quadratic convergence of the method under the constant rank constraint qualification, which is weaker than the frequently used linear independence constraint qualification, and a suitable second-order condition. Some numerical results are presented to illustrate the performance of the method.     

Implications of the constant rank constraint qualification

This paper investigates properties of a parametric set defined by finitely many equality and inequality constraints under the constant rank constraint qualification (CRCQ). We show, under the CRCQ, that the indicator function of this set is prox-regular with compatible parametrization, that the set-valued map that assigns each parameter to the set defined by that parameter satisfies a continuity property similar to the Aubin property, and that the Euclidean projector onto this set is a piecewise smooth function. We also show in the absence of parameters that the CRCQ implies the Mangasarian-Fromovitz constraint qualification to hold in some alternative expression of the set.     

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.     

Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications

We consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints. Problems of this kind arise in various applications including optimal topology design problems of mechanical structures. We show that some standard constraint qualifications like LICQ and MFCQ usually do not hold at a local minimum of our program, whereas the Abadie constraint qualification is sometimes satisfied. We also introduce a suitable modification of the standard Abadie constraint qualification as well as a corresponding optimality condition, and show that this modified constraint qualification holds under fairly mild assumptions. We also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints.     

Robust Duality in Parametric Convex Optimization

Modelling of convex optimization in the face of data uncertainty often gives rise to families of parametric convex optimization problems. This motivates us to present, in this paper, a duality framework for a family of parametric convex optimization problems. By employing conjugate analysis, we present robust duality for the family of parametric problems by establishing strong duality between associated dual pair. We first show that robust duality holds whenever a constraint qualification holds. We then show that this constraint qualification is also necessary for robust duality in the sense that the constraint qualification holds if and only if robust duality holds for every linear perturbation of the objective function. As an application, we obtain a robust duality theorem for the best approximation problems with constraint data uncertainty under a strict feasibility condition.     

Enhanced Karush–Kuhn–Tucker Conditions for Mathematical Programs with Equilibrium Constraints

In this paper, we study necessary optimality conditions for nonsmooth mathematical programs with equilibrium constraints. We first show that, unlike the smooth case, the mathematical program with equilibrium constraints linear independent constraint qualification is not a constraint qualification for the strong stationary condition when the objective function is nonsmooth. We then focus on the study of the enhanced version of the Mordukhovich stationary condition, which is a weaker optimality condition than the strong stationary condition. We introduce the quasi-normality and several other new constraint qualifications and show that the enhanced Mordukhovich stationary condition holds under them. Finally, we prove that quasi-normality with regularity implies the existence of a local error bound.     

Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow-Hurwicz-Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn-Tucker constraint qualification, and MPEC Abadie constraint qualification.     

Global and local convergence of a nonmonotone SQP method for constrained nonlinear optimization

In this paper, we propose a robust sequential quadratic programming (SQP) method for nonlinear programming without using any explicit penalty function and filter. The method embeds the modified QP subproblem proposed by Burke and Han (Math Program 43:277–303, 1989) for the search direction, which overcomes the common difficulty in the traditional SQP methods, namely the inconsistency of the quadratic programming subproblems. A non-monotonic technique is employed further in a framework in which the trial point is accepted whenever there is a sufficient relaxed reduction of the objective function or the constraint violation function. A forcing sequence possibly tending to zero is introduced to control the constraint violation dynamically, which is able to prevent the constraint violation from over-relaxing and plays a crucial role in global convergence and the local fast convergence as well. We prove that the method converges globally without the Mangasarian–Fromovitz constraint qualification (MFCQ). In particular, we show that any feasible limit point that satisfies the relaxed constant positive linear dependence constraint qualification is also a Karush–Kuhn–Tucker point. Under the strict MFCQ and the second order sufficient condition, furthermore, we establish the superlinear convergence. Preliminary numerical results show the efficiency of our method.     

Some theoretical limitations of second-order algorithms for smooth constrained optimization

In second-order algorithms, we investigate the relevance of the constant rank of the full set of active constraints in ensuring global convergence to a second-order stationary point under a constraint qualification. We show that second-order stationarity is not expected in the non-constant rank case if the growth of so-called tangent AKKT2 sequences is not controlled. Since no algorithm controls their growth, we argue that there is a theoretical limitation of algorithms in finding second-order stationary points without constant rank assumptions.     

A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints

In this paper, we consider a mathematical program with complementarity constraints. We present a modified relaxed program for this problem, which involves less constraints than the relaxation scheme studied by Scholtes (2000). We show that the linear independence constraint qualification holds for the new relaxed problem under some mild conditions. We also consider a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is C-stationary to the original problem under the MPEC linear independence constraint qualification and, if the Hessian matrices of the Lagrangian functions of the relaxed problems are uniformly bounded below on the corresponding tangent space, it is M-stationary. We also obtain some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices mentioned above are new and can be verified easily. This work was supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science, Sports, and Culture of Japan. The authors are grateful to an anonymous referee for critical comments.     

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.     

On the Guignard constraint qualification for mathematical programs with equilibrium constraints

We recapitulate the well-known fact that most of the standard constraint qualifications are violated for mathematical programs with equilibrium constraints (MPECs). We go on to show that the Abadie constraint qualification is only satisfied in fairly restrictive circumstances. In order to avoid this problem, we fall back on the Guignard constraint qualification (GCQ). We examine its general properties and clarify the position it occupies in the context of MPECs. We show that strong stationarity is a necessary optimality condition under GCQ. Also, we present several sufficient conditions for GCQ, showing that it is usually satisfied for MPECs.     

Global convergence of a robust filter SQP algorithm

We present a robust filter SQP algorithm for solving constrained optimization problems. This algorithm is based on the modified quadratic programming proposed by Burke to avoid the infeasibility of the quadratic programming subproblem at each iteration. Compared with other filter SQP algorithms, our algorithm does not require any restoration phase procedure which may spend a large amount of computation. The main advantage of our algorithm is that it is globally convergent without requiring strong constraint qualifications, such as Mangasarian–Fromovitz constraint qualification (MFCQ) and the constant rank constraint qualification (CRCQ). Furthermore, the feasible limit points of the sequence generated by our algorithm are proven to be the KKT points if some weaker conditions are satisfied. Numerical results are also presented to show the efficiency of the algorithm.