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.     

非线性互补约束优化问题的可行性条件

本文研究了非线性互补约束优化问题的可行性条件，其中约束条件除互补问题外还包括第一水平(设计)变量和第二水平(状态)变量同时出现的其它非线性约束，它是线性互补约束优化问题的可行性条件的推广。     

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.     

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.     

A New Extreme Point Algorithm and Its Application in PSQP Algorithms for Solving Mathematical Programs with Linear Complementarity Constraints

In this paper, we present a new extreme point algorithm to solve a mathematical program with linear complementarity constraints without requiring the upper level objective function of the problem to be concave. Furthermore, we introduce this extreme point algorithm into piecewise sequential quadratic programming (PSQP) algorithms. Numerical experiments show that the new algorithm is efficient in practice.     

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.     

Complementarity Active-Set Algorithm for Mathematical Programming Problems with Equilibrium Constraints

In this paper, an algorithm for solving a mathematical programming problem with complementarity (or equilibrium) constraints (MPEC) is introduced, which uses the active-set methodology while maintaining the complementarity restrictions throughout the procedure. Finite convergence of the algorithm to a strongly stationary point of the MPEC is established under reasonable hypotheses. The algorithm can be easily implemented by adopting any active-set code for nonlinear programming. Computational experience is included to highlight the efficacy of the proposed method in practice.     

Existence Theorems for Bilevel Problem with Applications to Mathematical Program with Equilibrium Constraint and Semi-Infinite Problem

In this paper, we establish existence theorems for bilevel problems with fixed-point constraints and bilevel problems without fixed-point constraint. The aim of this paper is to investigate under which conditions the existence of a feasible point of a bilevel problem can be assumed in advance and under which conditions there exist minimizers for this type of problems. From this, we establish existence theorems for mathematical programs with equilibrium constraints and semi-infinite problems. This research was supported by the National Science Council of the Republic of China. The author thanks the referees for suggestions and comments leading to the present form of the paper.     

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     

Superlinear Convergence of a Smooth Approximation Method for Mathematical Programs with Nonlinear Complementarity Constraints

<正>Mathematical programs with complementarity constraints(MPCC) is an important subclass of MPEC.It is a natural way to solve MPCC by constructing a suitable approximation of the primal problem.In this paper,we propose a new smoothing method for MPCC by using the aggregation technique.A new SQP algorithm for solving the MPCC problem is presented.At each iteration,the master direction is computed by solving a quadratic program,and the revised direction for avoiding the Maratos effect is generated by an explicit formula.As the non-degeneracy condition holds and the smoothing parameter tends to zero,the proposed SQP algorithm converges globally to an S-stationary point of the MPEC problem,its convergence rate is superlinear.Some preliminary numerical results are reported.     

非线性互补约束规划问题的一个新的QP-free算法

本文研究了非线性互补约束均衡问题.利用互补函数以及光滑近似法,把非线性互补约束均衡问题转化为一个光滑非线性规划问题,得到了超线性收敛速度,数值实验结果表明本文提出的算法是可行的.     

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     

Active set algorithm for mathematical programs with linear complementarity constraints

In the paper, an incomplete active set algorithm is given for mathematical programs with linear complementarity constraints (MPLCC). At each iteration, a finite number of inner-iterations are contained for approximately solving the relaxed nonlinear optimization problem. If the feasible region of the MPLCC is bounded, under the uniform linear independence constraint qualification (LICQ), any cluster point of the sequence generated from the algorithm is a B-stationary point of the MPLCC. Preliminary numerical tests show that the algorithm is promising.     

Multiplier convergence in trust-region methods with application to convergence of decomposition methods for MPECs

We study piecewise decomposition methods for mathematical programs with equilibrium constraints (MPECs) for which all constraint functions are linear. At each iteration of a decomposition method, one step of a nonlinear programming scheme is applied to one piece of the MPEC to obtain the next iterate. Our goal is to understand global convergence to B-stationary points of these methods when the embedded nonlinear programming solver is a trust-region scheme, and the selection of pieces is determined using multipliers generated by solving the trust-region subproblem. To this end we study global convergence of a linear trust-region scheme for linearly-constrained NLPs that we call a trust-search method. The trust-search has two features that are critical to global convergence of decomposition methods for MPECs: a robustness property with respect to switching pieces, and a multiplier convergence result that appears to be quite new for trust-region methods. These combine to clarify and strengthen global convergence of decomposition methods without resorting either to additional conditions such as eventual inactivity of the trust-region constraint, or more complex methods that require a separate subproblem for multiplier estimation.      

Mathematical Programs with Vector Optimization Constraints

In this paper, we introduce mathematical programs with vector optimization constraints. For these problems, we establish two models in both the weak Pareto solution and Pareto solution setting. Some new existence results are obtained under rather weak conditions. We establish also equivalences between mathematical programs with vector optimization constraints and mathematical programs with vector variational inequality constraints.This research was partially supported by a grant from the National Science Council of the ROC. The authors thank the referees for helpful suggestions and comments.     

Convergence of an Inexact Smoothing Method for Mathematical Programs with Equilibrium Constraints

In this paper, we propose an inexact smoothing continuation method for mathematical problem with complementarity constraints. Under suitable conditions, we establish the convergence of the proposed method by showing that any accumulation point of the generated sequence is a B-stationary point of the problem.     

隐互补约束优化问题的磨光SQP算法

本文提出了一类隐互补约束优化问题的磨光SQP算法.首先,我们给出了这类优化问题的最优性和约束规范性条件.然后,在适当假设条件下,我们证明了算法具有全局收敛性.     

A Robust SQP Method for Mathematical Programs with Linear Complementarity Constraints

The relationship between the mathematical program with linear complementarity constraints (MPLCC) and its inequality relaxation is studied. Based on this relationship, a new sequential quadratic programming (SQP) method is presented for solving the MPLCC. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results are derived without assuming the linear independence constraint qualification for MPEC, the nondegeneracy condition, and any feasibility condition of the quadratic programming subproblems. Preliminary numerical results are reported. Research is partially supported by Singapore-MIT Alliance and School of Business, National University of Singapore.     

QPECgen, a MATLAB Generator for Mathematical Programs with Quadratic Objectives and Affine Variational Inequality Constraints

We describe a technique for generating a special class, called QPEC, of mathematical programs with equilibrium constraints, MPEC. A QPEC is a quadratic MPEC, that is an optimization problem whose objective function is quadratic, first-level constraints are linear, and second-level (equilibrium) constraints are given by a parametric affine variational inequality or one of its specialisations. The generator, written in MATLAB, allows the user to control different properties of the QPEC and its solution. Options include the proportion of degenerate constraints in both the first and second level, ill-conditioning, convexity of the objective, monotonicity and symmetry of the second-level problem, and so on. We believe these properties may substantially effect efficiency of existing methods for MPEC, and illustrate this numerically by applying several methods to generator test problems. Documentation and relevant codes can be found by visiting http://www.ms.unimelb.edu.au/danny/qpecgendoc.html.     

Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints

Using the theory of exact penalization for mathematical programs with subanalytic constraints, the theory of error bounds for quadratic inequality systems, and the theory of parametric normal equations, we derive various exact penalty functions for mathematical programs subject to equilibrium constraints, and we also characterize stationary points of these programs. The research of this author is based on work supported by the National Sciences and Engineering Research Council of Canada under grant OPG0090391. The research of this author is based on work supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739. Part of this paper was completed while he was visiting The University of Melbourne and The University of New South Wales. The research of this author is based on work supported by the Australian Research Council.