解互补约束优化问题的一种新的光滑化近似方法

A new smoothing method for mathematical programs with complementarity constraints

互补约束优化问题应用十分广泛.利用Sigmoid函数的积分函数提出了一种新的光滑化近似算法,将互补约束优化问题转化为一般的非线性规划近似问题,通过求解近似问题的一系列光滑子问题得到原问题的近似解.在线性独立约束规范和其他一些较弱的假设条件下:无须上水平严格互补和渐进弱非退化,证明了光滑近似问题的KKT稳定点序列收敛于原问题的C-稳定点.进而考虑弱二阶必要条件,证明了上述KKT稳定点序列收敛于原问题的S-稳定点.最后,设计了相应算法,并对MacMPEC测试题库中的一些算例进行了数值实验,将得到的结果与其他算法的结果进行比较,显示本方法是有效的.

Mathematical programs with complementarity constraints(MPCC)have very wide application in many areas.In this paper,we present a smoothing method based on the integral of the Sigmoid function.The original MPCC is reformulated into a standard smooth optimization model.Then,we obtain an approximate solution of MPCC by solving a series of the smooth sub-problems.It is proved that any accumulation point of the KKT stationary points sequence is a C-stationary point of original MPCC under the linear independence constraints qualification(LICQ)and the other weaker assumptions,without upper level strict complementarity(ULSC)and asymptotically weakly nondegenerate condition(AWN).Furthermore,such an accumulation point can be proved to be S-stationary point under the weak second-order necessary condition.At last,we present the algorithm and test its efficiency with some problems in MacMPEC database.Numerical results indicate that the proposed smoothing method is efficient comparing with other related algorithms.