不可微规划的二阶最优性条件

Second order optimality conditions for nondifferentiable programming

研究了一类复合不可微规划：ｍｉｎｘ∈ＲｎＦ（ｘ），其中Ｆ∶＝ｈｆ，ｈ：Ｒｍ→Ｒ是凸函数，ｆ：Ｒｎ→Ｒｍ是Ｃ１，１函数．给出了其二阶最优性条件：（ｉ）若Ｆ在ｚ处取局部极小，则对ｄ∈Ｋ（ｚ），有ｍａｘｙ＊∈Ｍ（ｚ）｛ｄＴＡｄ｜Ａ∈２ｘｘＬ（ｚ，ｙ＊）｝≥０；（ｉ）若Ｍ（ｚ）≠，且对ｄ∈Ｄ（ｚ），ｍａｘｙ＊∈Ｍ（ｚ）｛ｄＴＡｄ｜Ａ∈２ｘｘＬ（ｚ，ｙ＊）｝＞０。

The problems are studied: min x ∈R nF(x) ,where F∶=hf with h :R m→R is convex and F :R n→R m is C 1,1 function,and gives the second optimality conditions: (i) If F attains a local minimumat z . then max y ∈M(z) {d TAd|A∈ 2 xx L(z,y )} ≥0 whenever d∈ k(z) ;(ii) If z ∈R n is such that M(z) ≠ and max y ∈M(z){d TAd|A∈ 2 xx L(z,y ) }>0,for all D∈(z) ,then Z is an isolated local minimum for F. The results generalize the theory developed by J.V.Burke.