Banach空间中一类扰动优化问题最优解的特征与存在性

设(X,‖·‖)是Banach空间,x∈X,Z是X的非空子集,J是Z→R的下半连续下有界函数．本文研究扰动优化问题min_(z∈Z)(J(z)+‖x-z‖)(记作(J,x)-inf)的最优解的特征和最优解的存在性等问题．我们引入J－太阳集的概念,同时在Z是J-太阳集的情形下,给出了扰动优化问题(J,x)-inf的最优解的“Kolmogorov”型特征刻画．并借助于集合的若干紧性概念和最优值函数的方向导数研究了扰动优化问题(J,x)-inf的最优解的存在性．

Characterization and Existence of Optimal Solution to a Class of Perturbed Optimization Problems in Banach Spaces

Let (X,‖·‖) be a Banach space and Z be a nonempty subset of X.Let J:Z→R be a lower semicontinuous function bounded from below.The present paper concerns the problems of the characterization and the existence of the optimal solution of the perturbed optimization problem min_(zeZ)(J(z)+‖x-z‖),denoted by (J,x)-inf.The new notion of the J-sun set is introduced and the characterizations of the optimal solution of the problem (J,x)-inf for J-suns are provided in terms of Kolmogorov Criterion.Moreover,some existence results of the solution of the problem (J,x)-inf are given with the aid of some compactness of sets or the directional derivative of the optimal-value functional.