一类非光滑规划问题的最优性条件

Optimality Conditions of a Class of Nonsmooth Programming Problem

本文给出了带等式和不等式约束的非光滑B-(p,r)规划问题的KKT必要性条件,即:若∈D是(P)的最优解,∑mi=1μigi+∑pj=1vjhj在处是关于η和b的严格B-(p,r)不变凸函数,gi(i∈I),hj(j∈J1),-hj(j∈J2)在处正则。则存在λ>0,μ∈Rm+,v∈Rp,使得是(P)的KKT点。同时,也给出了该类规划问题的KKT充分条件,即:若∈D处KKT条件(2)～(4)式,f+∑mi=1μigi+∑pj=1vjhj在处是关于η和b的B-(p,r)不变凸函数且f,gi(i∈I),hj(j∈J1),-hj(j∈J2)在处正则,那么是(P)的最优解。

In this paper necessary KKT condition is given to a class of nonsmooth B-（p, r） programming problems with equality and P inequality constraints as follows： Let x∈ D be optimal solution for （P）, ∑i=1^mμigi＋∑j=1^pvjhj is strictly B-（p, r） invex function at with respect to η and b, gi（i∈1）,hj（j∈J1）,-hj（j∈J2） are regular at x. Then there exist λ 〉 0 ,μ∈R＋^m,v∈R^p, such that x is KKT point for （P）. At the same time, the sufficient KKT condition is given to this programming problem as follows：Let KKT conditions （2） - （4） are satisfied at x∈D, f＋∑i=1^μigi＋∑j=1^pvjhj is B- （ p, r） invex function at x with respect to -η and b, gi （ i ∈ I）, hj（j∈J1）, - hi（j ∈ J2 ） are regular at x. Then x is an optimal solution for （P）.