有效解的—阶广义梯度条件  

One Order Generalized Gradient Conditions of Efficient Solutions

作　　者：李雨生[1] 倪明放[2] 

机构地区：[1]河海大学数理系 [2]南京通信工程学院

出　　处：《河海大学学报（自然科学版）》1990年第1期76-81,共6页Journal of Hohai University(Natural Sciences)

摘　　要：本文讨论带闭凸锥的多目标优化问题.设f(x)是目标向量函数,g(x)是约束向量函数,M, -N分别是它们的控制锥.当x是弱有效解,则?;当x是绝对有效解,则▽f(x)是零矩阵.而当f(x)是M-凸函数,g(x)是N-拟凸函数,则存在λ,使0∈?(x^rf)(x).这里对应于x是有效解和Hartley真有效解分别有λ∈M·\{0}和λ∈intM.M表示M的正极锥, 表Clarke广义梯度集.而锥拟凸函数是我们提出的一种比锥凸函数更广泛的函数,我们称g(x)是N—拟凸的是指对R^m中的任何α,{x∈X|g(x)≤N~α}是凸集.另外,当x是Hortley真有效解,还存在m×k阶矩阵,使0∈?[λ~r(f+Ag)(x)],而λ∈M·|{0}.This paper discusses multiobjective optimization with closed convex cones. The main results obtained are as followings: f (x) is taken as the objective vector function, and g (x) as the subject vector function, M, —N as their dominating cones respectively. When f (x) is M—convex and g (x) is N—quasiconvex, then there is vector λ to make zero an element of Clarke generalized gradient set of (λ~Tf)(x) at ?. If ? is efficient solution or Hartley proper efficient solution, then λ∈EM\ {0} or λ∈ intM. Correspondingly, for ? being the weak efficient, ?f (?) ? intM, and for ? being the absolute efficient, ▽f (?) is zero matrix, and for ? being the Hartley properly efficient, there is an m×k matrix A to make zero an element of Clarke generalized gradient set of λ~T (f+Ag) (x) at ?. The cone quasiconvexity proposed here is a weaker condition than that of cone convexity. If {x∈X|g (x) ≤Nα} i a convex set for any vector α in R^m, we call g (x) an N—quasiconvex function.

关 键 词：凸函数 一阶 广义 梯度条件 

分 类 号：O174.13[理学—数学]