带有互补约束的数学规划(MPCC)问题是一类难于求解的优化问题,其在许多领域都有着重要的应用。针对互补约束的特殊结构,人们提出了多种方法求解MPCC问题。近年来,非线性优化问题的序列最优性条件受到了广泛的关注。基于序列最优性条件,...带有互补约束的数学规划(MPCC)问题是一类难于求解的优化问题,其在许多领域都有着重要的应用。针对互补约束的特殊结构,人们提出了多种方法求解MPCC问题。近年来,非线性优化问题的序列最优性条件受到了广泛的关注。基于序列最优性条件,算法的收敛性结果得到了显著改进。但是,非线性优化问题的序列最优性条件不能直接用于研究MPCC问题。因此,本文基于非线性优化问题(NLP)的CAKKT条件,提出了MPCC问题关于M稳定性的序列最优性条件,即MPCC-CAKKT条件。MPCC-CAKKT条件是比现有的MPCC-AKKT条件更强的序列最优性条件。此外,还给出了与之相关的保证M稳定性的较弱的约束规范,即MPCC-CAKKT正则性。Mathematical program with complementarity constraints (MPCC) is a difficult class of optimization problems, which plays an important role in many fields. Due to the special structure of the complementarity constraints, several methods have been suggested in order to deal with the MPCC. Recently, the sequential optimality conditions for nonlinear optimization problems (NLP) have been drawn concerns widely. Convergence analysis of these methods for NLP has been dramatically improved by using the sequential optimality conditions. However, the established sequential optimality conditions for NLP are not suitable for MPCC. In this paper, based on the CAKKT condition for NLP, we present a sequential optimality condition for MPCC, namely MPCC-CAKKT condition, which is stronger than the MPCC-AKKT condition. Furthermore, we present a weaker constraint qualification for M stationarity which is closely related to MPCC-CAKKT.展开更多
文摘带有互补约束的数学规划(MPCC)问题是一类难于求解的优化问题,其在许多领域都有着重要的应用。针对互补约束的特殊结构,人们提出了多种方法求解MPCC问题。近年来,非线性优化问题的序列最优性条件受到了广泛的关注。基于序列最优性条件,算法的收敛性结果得到了显著改进。但是,非线性优化问题的序列最优性条件不能直接用于研究MPCC问题。因此,本文基于非线性优化问题(NLP)的CAKKT条件,提出了MPCC问题关于M稳定性的序列最优性条件,即MPCC-CAKKT条件。MPCC-CAKKT条件是比现有的MPCC-AKKT条件更强的序列最优性条件。此外,还给出了与之相关的保证M稳定性的较弱的约束规范,即MPCC-CAKKT正则性。Mathematical program with complementarity constraints (MPCC) is a difficult class of optimization problems, which plays an important role in many fields. Due to the special structure of the complementarity constraints, several methods have been suggested in order to deal with the MPCC. Recently, the sequential optimality conditions for nonlinear optimization problems (NLP) have been drawn concerns widely. Convergence analysis of these methods for NLP has been dramatically improved by using the sequential optimality conditions. However, the established sequential optimality conditions for NLP are not suitable for MPCC. In this paper, based on the CAKKT condition for NLP, we present a sequential optimality condition for MPCC, namely MPCC-CAKKT condition, which is stronger than the MPCC-AKKT condition. Furthermore, we present a weaker constraint qualification for M stationarity which is closely related to MPCC-CAKKT.
