ABSTRACT

Mathematical programs with equilibrium constraints is a difficult class of constrained optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications. Thus, the Karush–Kuhn–Tucker conditions are not necessarily satisfied at minimizers, and the convergence assumptions of many methods for solving constrained optimization problems are not fulfilled. Thus, it is necessary, from a theoretical and numerical point of view, to consider suitable optimality conditions, tailored constraints qualifications, and designed algorithms for solving such optimization problems. In this paper, we present a new sequential optimality condition useful for the convergence analysis of several methods for solving mathematical programs with equilibrium constraints such as relaxations schemes, complementarity-penalty methods, and interior-relaxation methods. Furthermore, the weakest constraint qualification for M-stationarity associated with such sequential optimality condition is presented. Relations between the old and new constraint qualifications, as well as the algorithmic consequences, will be discussed.

摘要

具有平衡约束的数学程序是一类困难的约束优化问题。可行集具有非常特殊的结构，并且违反了大多数标准约束条件。因此，最小化器不一定满足Karush–Kuhn–Tucker条件，并且不能满足许多​​解决约束优化问题的方法的收敛假设。因此，有必要从理论和数值的角度考虑合适的最优性条件，量身定制的约束条件以及设计用于解决此类优化问题的算法。在本文中，我们提出了一种新的顺序最优性条件，可用于几种求解具有平衡约束的数学程序的方法（例如松弛方案，互补惩罚法和内部放松法。此外，提出了与这种顺序最优性条件相关的M平稳性的最弱约束条件。将讨论新旧约束资格之间的关系以及算法的后果。