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.     

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.     

A Class of Quadratic Programs with Linear Complementarity Constraints

We consider a class of quadratic programs with linear complementarity constraints (QPLCC) which belong to mathematical programs with equilibrium constraints (MPEC). We investigate various stationary conditions and present new and strong necessary and sufficient conditions for global and local optimality. Furthermore, we propose a Newton-like method to find an M-stationary point in finite steps without MEPC linear independence constraint qualification. The research of this author is partially supported by NSERC, and Research Grand Council of Hong Kong.     

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.      

Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints

Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primal-dual interior-point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced under fairly general conditions other than strict complementarity or the linear independence constraint qualification for MPEC (MPEC-LICQ). It is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a point with certain stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interior-point algorithm failed to find a stationary point.Mathematics Subject Classification (1991):90C30, 90C33, 90C55, 49M37, 65K10     

Convergence properties of the inexact Lin-Fukushima relaxation method for mathematical programs with complementarity constraints

Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, form a difficult class of optimization problems. The feasible set of MPECs is described by standard equality and inequality constraints as well as additional complementarity constraints that are used to model equilibrium conditions in different applications. But these complementarity constraints imply that MPECs violate most of the standard constraint qualifications. Therefore, more specialized algorithms are typically applied to MPECs that take into account the particular structure of the complementarity constraints. One popular class of these specialized algorithms are the relaxation (or regularization) methods. They replace the MPEC by a sequence of nonlinear programs NLP(t) depending on a parameter t, then compute a KKT-point of each NLP(t), and try to get a suitable stationary point of the original MPEC in the limit t→0. For most relaxation methods, one can show that a C-stationary point is obtained in this way, a few others even get M-stationary points, which is a stronger property. So far, however, these results have been obtained under the assumption that one is able to compute exact KKT-points of each NLP(t). But this assumption is not implementable, hence a natural question is: What kind of stationarity do we get if we only compute approximate KKT-points? It turns out that most relaxation methods only get a weakly stationary point under this assumption, while in this paper, we show that the smooth relaxation method by Lin and Fukushima (Ann. Oper. Res. 133:63–84, 2005) still yields a C-stationary point, i.e. the inexact version of this relaxation scheme has the same convergence properties as the exact counterpart.     

A linear programming proof of the second order conditions of non-linear programming

In this note we give a new, simple proof of the standard first and second order necessary conditions, under the Mangasarian–Fromovitz constraint qualification (MFCQ), for non-linear programming problems. We work under a mild constraint qualification, which is implied by MFCQ. This makes it possible to reduce the proof to the relatively easy case of inequality constraints only under MFCQ. This reduction makes use of relaxation of inequality constraints and it makes use of a penalty function. The new proof is based on the duality theorem for linear programming; the proofs in the literature are based on results of mathematical analysis. This paper completes the work in a recent note of Birbil et al. where a linear programming proof of the first order necessary conditions has been given, using relaxation of equality constraints.     

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.     

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.     

On M-stationary points for mathematical programs with equilibrium constraints

Mathematical programs with equilibrium constraints are optimization problems which violate most of the standard constraint qualifications. Hence the usual Karush-Kuhn-Tucker conditions cannot be viewed as first order optimality conditions unless relatively strong assumptions are satisfied. This observation has lead to a number of weaker first order conditions, with M-stationarity being the strongest among these weaker conditions. Here we show that M-stationarity is a first order optimality condition under a very weak Abadie-type constraint qualification. Our approach is inspired by the methodology employed by Jane Ye, who proved the same result using results from optimization problems with variational inequality constraints. In the course of our investigation, several concepts are translated to an MPEC setting, yielding in particular a very strong exact penalization result.     

Optimality conditions for nonsmooth mathematical programs with equilibrium constraints,using convexificators

In this paper, we deal with constraint qualifications, stationary concepts and optimality conditions for a nonsmooth mathematical program with equilibrium constraints (MPEC). The main tool in our study is the notion of convexificator. Using this notion, standard and MPEC Abadie and several other constraint qualifications are proposed and a comparison between them is presented. We also define nonsmooth stationary conditions based on the convexificators. In particular, we show that GS-stationary is the first-order optimality condition under generalized standard Abadie constraint qualification. Finally, sufficient conditions for global or local optimality are derived under some MPEC generalized convexity assumptions.     

Merit-Function Piecewise SQP Algorithm for Mathematical Programs with Equilibrium Constraints

We propose a merit-function piecewise SQP algorithm for mathematical programs with equilibrium constraints (MPEC) formulated as mathematical programs with complementarity constraints. Under mild conditions, the new algorithm is globally convergent to a piecewise stationary point. Moreover, if the partial MPEC linear independence constraint qualification (LICQ) is satisfied at the accumulation point, then the accumulation point is an S-stationary point. The research of the first author was supported by the National Natural Science Foundation of China under grants 10571177 and 70271014. The research of the second author was partially supported by NSERC.     

ABS算法在求解MPEC问题中的应用

应用ABS—隐式LU算法，简化MPEC问题的约束条件，将简化后的MPEC问题转化为目标函数带有罚函数子项的非线性无约束优化问题，给出收敛性定理，证明当罚因子足够大时，此非线性无约束问题的极小点就是简化后的MPEC问题的极小点，将此极小点代入本中给出的一个转换公式可得原MPEC问题的极小点，末给出一算例。     

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.     

On constraint qualifications

The linear independence constraint qualification (LICQ) and the weaker Mangasarian-Fromovitz constraint qualification (MFCQ) are well-known concepts in nonlinear optimization. A theorem is proved suggesting that the set of feasible points for which MFCQ essentially differs from LICQ is small in a specified sense. As an auxiliary result, it is shown that, under MFCQ, the constraint set (even in semi-infinite optimization) is locally representable in epigraph form.The author wishes to thank Professor H. T. Jongen for valuable advice.     

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.     

A Gauss–Newton Approach for Solving Constrained Optimization Problems Using Differentiable Exact Penalties

We propose a Gauss–Newton-type method for nonlinear constrained optimization using the exact penalty introduced recently by André and Silva for variational inequalities. We extend their penalty function to both equality and inequality constraints using a weak regularity assumption, and as a result, we obtain a continuously differentiable exact penalty function and a new reformulation of the KKT conditions as a system of equations. Such reformulation allows the use of a semismooth Newton method, so that local superlinear convergence rate can be proved under an assumption weaker than the usual strong second-order sufficient condition and without requiring strict complementarity. Besides, we note that the exact penalty function can be used to globalize the method. We conclude with some numerical experiments using the collection of test problems CUTE.     

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.