Mathematical programs with vanishing constraints: a new regularization approach with strong convergence properties

Motivated by a recent method introduced by Kanzow and Schwartz [C. Kanzow and A. Schwartz, A new regularization method for mathematical programs with complementarity constraints with strong convergence properties, Preprint 296, Institute of Mathematics, University of Würzburg, Würzburg, 2010] for mathematical programs with complementarity constraints (MPCCs), we present a related regularization scheme for the solution of mathematical programs with vanishing constraints (MPVCs). This new regularization method has stronger convergence properties than the existing ones. In particular, it is shown that every limit point is at least M-stationary under a linear independence-type constraint qualification. If, in addition, an asymptotic weak nondegeneracy assumption holds, the limit point is shown to be S-stationary. Second-order conditions are not needed to obtain these results. Furthermore, some results are given which state that the regularized subproblems satisfy suitable standard constraint qualifications such that the existing software can be applied to these regularized problems.     

First-and second-order optimality conditions for mathematical programs with vanishing constraints

We consider a special class of optimization problems that we call Mathematical Programs with Vanishing Constraints, MPVC for short, which serves as a unified framework for several applications in structural and topology optimization. Since an MPVC most often violates stronger standard constraint qualification, first-order necessary optimality conditions, weaker than the standard KKT-conditions, were recently investigated in depth. This paper enlarges the set of optimality criteria by stating first-order sufficient and second-order necessary and sufficient optimality conditions for MPVCs. Dedicated to Jiří V. Outrata on the occasion of his 60th birthday. This research was partially supported by the DFG (Deutsche Forschungsgemeinschaft) under grant KA1296/15-1.     

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.     

Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications

We consider a class of optimization problems that is called a mathematical program with vanishing constraints (MPVC for short). This class has some similarities to mathematical programs with equilibrium constraints (MPECs for short), and typically violates standard constraint qualifications, hence the well-known Karush-Kuhn-Tucker conditions do not provide necessary optimality criteria. In order to obtain reasonable first order conditions under very weak assumptions, we introduce several MPVC-tailored constraint qualifications, discuss their relation, and prove an optimality condition which may be viewed as the counterpart of what is called M-stationarity in the MPEC-field.     

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.     

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     

A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints

This paper presents a sequential quadratic programming algorithm for computing a stationary point of a mathematical program with linear complementarity constraints. The algorithm is based on a reformulation of the complementarity condition as a system of semismooth equations by means of Fischer-Burmeister functional, combined with a classical penalty function method for solving constrained optimization problems. Global convergence of the algorithm is established under appropriate assumptions. Some preliminary computational results are reported.     

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。     

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     

遗传算法求解约束非线性规划及Matlab实现

对于约束非线性规划问题,传统的方法:可行方向法、惩罚函数法计算烦琐且精度不高.用新兴的遗传算法来解决约束非线性规划,核心是惩罚函数的构造.以前的惩罚函数遗传算法有的精度较低,有的过于复杂.本文在两个定义的基础上构造了新的惩罚函数,并在新的惩罚函数的基础上,提出了一种解决约束非线性最优化问题的方法.通过两个例子应用Matlab说明了这个算法的可行性.     

On Some Feasibility Conditions in MPEC

This article discusses feasibility conditions in mathematical programs with equilibrium constraints （MPECs）. The authors prove that two sufficient conditions guarantee the feasibility of these MPECs. The authors show that the two feasibility conditions are different from the feasibility condition in [2, 3], and show that the sufficient condition in [3] is stronger than that in [2].     

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.     

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.     

互补约束数学规划问题的一个广义梯度投影罚算法

结合罚函数思想和广义梯度投影技术,提出求解非线性互补约束数学规划问题的一个广义梯度投影罚算法.首先,通过扰动技术和广义互补函数,将原问题转化为序列带参数的近似的标准非线性规划;其次,利用广义梯度投影矩阵构造搜索方向的显式表达式.一个特殊的罚函数作为效益函数,而且搜索方向能保证效益函数的下降性.在适当的假设条件下算法具有全局收敛性.     

Characterizations of linear suboptimality for mathematical programs with equilibrium constraints

The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasivariational inequalities, implicit complementarity problems, etc. Research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451168.     

互补约束规划问题的一个广义梯度投影算法

本文研究了一类均衡约束最优化问题.利用广义梯度投影法,结合罚函数思想,得到了一个初始点可以任意的广义梯度投影算法.在较弱的条件下,证明了算法的全局收敛性.     

A log-exponential smoothing method for mathematical programs with complementarity constraints

In this paper a log-exponential smoothing method for mathematical programs with complementarity constraints (MPCC) is analyzed, with some new interesting properties and convergence results provided. It is shown that the stationary points of the resulting smoothed problem converge to the strongly stationary point of MPCC, under the linear independence constraint qualification (LICQ), the weak second-order necessary condition (WSONC), and some reasonable assumption. Moreover, the limit point satisfies the weak second-order necessary condition for MPCC. A notable fact is that the proposed convergence results do not restrict the complementarity constraint functions approach to zero at the same order of magnitude.     

An overview of bilevel optimization

This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints). B. Colson now at SAMTECH s.a., Liège, Belgium. This article is an updated version of a paper that appeared in 4OR 3, 87–107, 2005.     

不等式约束优化问题的精确罚函数法

1引言 在约束最优化的研究中，罚函数法有很高的理论及应用价值，为求约束优化问题的最优解x，很多方法是通过求解一系列优化问题来实现，人们称之为SUMT方法~[1].     

A globally convergent version of Robinson's algorithm for general nonlinear programming problems without using derivatives

Robinson's quadratically convergent algorithm for general nonlinear programming problems is modified in such a way that, instead of exact derivatives of the objective function and the constraints, approximations of these can be used which are computed by differences of function values. This locally convergent algorithm is then combined with a penalty function method to provide a globally and quadratically convergent algorithm that does not require the calculation of derivatives.