最优化与变分不等式的可行解序列的有限终止性

The Finite Termination of Feasible Solution Sequence for Optimization and Variational Inequality

为了在更弱的条件下,给出最优化问题(OP)与变分不等式问题(VIP)的可行解序列的有限终止性,在这类问题的解集上引进了一个增广映射,分别建立了解集关于可行解序列广义弱尖锐性的概念.这个新概念是传统的弱尖锐性与强非退化概念的扩充与推广,其克服了最优化与变分不等式在许多情况下解集不具有弱尖锐性或强非退化性的缺陷.在这些问题的解集满足广义弱尖锐性的条件下,提供其可行解序列有限终止于解集的充分与必要条件.这些结果是现有相关文献中在弱尖锐或强非退化条件下相应结果的推广,同时也为许多最优化算法的有限终止性提供了更弱的充分条件.

To provide a finite characterization of feasible solution sequences for optimization problems(OP)and variational inequality problems(VIP),an augmented set value map is introduced for the solution sets of these problems.Additionally,the concepts of augmented weak sharpness with respect to feasible solution sequences are established.These novel notions extend the traditional concepts of weak sharpness and strongly non-degeneracy relative to feasible solution sequences,addressing the limitation that solution sets often lack weak sharpness or strongly non-degeneracy in many cases.When the feasible solution sets of optimization problems and variational inequality problems exhibit augmented weak sharpness,the necessary and sufficient conditions for the finite termination of feasible solution sequences are provided for each problem.These conditions extend the corresponding results found in existing literature,where solution sets are weak sharp or strongly non-degenerate.Furthermore,sufficient conditions with fewer restrictions are provided for the finite termination of various optimization algorithms.