利用Greenberg-Pierskalla次微分研究半无限拟凸规划的Karush-Kuhn-Tucker条件

Karush-Kuhn-Tucker Conditions for Semi-infinite Quasiconvex Programming by Greenberg-Pierskalla Subdifferential

最优性条件是最优化理论的一个重要研究方向,为优化算法的研究提供了重要的理论基础,且近几十年来在凸规划中已经获得丰富的理论成果。然而,拟凸函数的性质导致KKT条件在拟凸规划中很少被表示,尤其是半无限情形。考虑半无限拟凸规划,利用Greenberg-Pierskalla次微分的并集的凸包来刻画约束集的法锥,得到相应的KKT充分必要条件。本文定理4和5将文献[1]中定理5拓展到半无限拟凸规划。最后将本文定理5应用到半无限凸规划情形。

As an important research direction of optimization theories,the optimality condition has provided an important theoretical basis for the research of optimization algorithm.In recent decades,abundant theoretical results have been achieved in convex programming.However,the properties of quasiconvex functions have led to the rare formulationg of KKT condition for quasiconvex programmings,especially for the semi-infinite case.Considering the semi-infinite quasiconvex programming,corresponding KKT sufficient and necessary conditions are obtained by characterizing the normal cone of the constraint set by the convex hull of the union of Greenberg-Pierskalla subdifferentials.Theorem 5 by Suzuki^([1])is extended to a semi-infinite quasiconvex programming by Theorems 4 and 5 proposed in this paper.At last,Theorem 5 is applied to the case of semi-infinite convex programming.