In this paper, a class of smoothing penalty functions is proposed for optimization problems with equality, inequality and bound constraints. It is proved exact, under the condition of weakly generalized Mangasarian–Fromovitz constraint qualification, in the sense that each local optimizer of the penalty function corresponds to a local optimizer of the original problem. Furthermore, necessary and sufficient conditions are discussed for the inverse proposition of exact penalization. Based on the theoretical results in this paper, a class of smoothing penalty algorithms with feasibility verification is presented. Theories on the penalty exactness, feasibility verification and global convergence of the proposed algorithm are presented. Numerical results show that this algorithm is effective for nonsmooth nonconvex constrained optimization problems.

本文针对具有等式，不等式和边界约束的优化问题，提出了一类平滑惩罚函数。在弱广义Mangasarian–Fromovitz约束条件的条件下，从罚函数的每个局部优化器对应于原始问题的局部优化器的意义上来说，证明是正确的。此外，讨论了精确惩罚的反命题的必要条件和充分条件。基于本文的理论结果，提出了一类具有可行性验证的平滑惩罚算法。提出了惩罚精确度，可行性验证和全局收敛的理论。数值结果表明，该算法对非光滑非凸约束优化问题有效。