摘要
本文首先介绍Rockafellar凸分析中回收锥、回收函数概念的提出,并将其中所介绍的回收锥、回收函数的性质进行归纳整理。
第二部分介绍回收锥、回收函数的某些理论及应用研究:
(1)研究了R~n中一般集合的广义回收锥和上一般函数的广义回收函数,推广了Rockafellar关于凸集和凸函数的回收锥和回收函数的一些结果。
(2)函数和集合的下降方向和可行方向之间的关系可以由其回收锥来刻画。进而讨论最优解存在性的局部最优性条件和全局最优性条件的关系。
(3)介绍一种不可微优化问题的光滑化方法。通过一个光滑参数的控制用一近似问题代替原问题。其中回收函数在构造近似问题中起重要作用。近似问题的最优解和原问题的最优解之间的差别可以通过光滑参数来调控。从而导致原近似问题和其相应的对偶问题之间的关系。
(4)利用回收函数讨论优化问题中无界性的数值方法。
(5)讨论无限维空间中非凸集合的回收锥,并将其结果应用到研究向量优化问题中的效率条件和优势性质中。
(6)给出一线性拓扑空间中其回收锥有非空内部的闭集的一局部刻画定理,继而用这一定理来刻画定义在E~d中的闭子集上的上半连续增函数。
第三部分将W.T.Obuchowska和K.B.Murty(2001)关于可微凸函数的回收锥的某些研究结果推广到一般凸函数的情形,其中最为重要的结果是一向量s为函数f的回收方向的充要条件。
In this article,we first introduce the conception of recession cone and recession function in convex analysis written by Rockafellar.And the propositions of recession cone and recession function in convex analysis are summarized.
In the second part,we introduce certain applications and some theoretical research of recession cone and recession function:
(1) We study generalized recession cone for general subset C of R~n and generalized recession function for general function.Some results for recession cone of convex set and recession function of convex function have been generalized.
(2) Recession cones of functions and sets are characterized as intersections of descent directions and feasible directions,respectively.Relations between local optimality conditions and global conditions for the existence of optimal points are then studied.
(3) We introduce a smoothing techniquefor nondifferentiable optimization problems. The approach is to replace the original problem by an approximate one which is controlled by a smoothing parameter.The recession function plays an important role in approximate problem.An a priori bound on the difference between the optimal values of the original problem and the approximate one is explicitly derived in term of the smoothing parameter. The relationships between the primal approximated problem and its corresponding dual are investigated.
(4) We use recession function to discuss numerical method of unbounded in optimization.
(5) Recession cones ofnonconvex sets in infinite dimensional spaces are studied The results are then applied to investigate efficiency conditions and the domination property in vector optimization.
(6) A local characterization theorem is given for closed sets in a linear topological space that have recession cones with nonempty interior.This theorem is then used to characterize the class of upper semicontinuous increasing functions defined on closed subsets of E~d.
In the third part,some important results of recession cone of convex functions are introduced in this paper by generalizing the results of W.T.Obuchowska for differentiable convex functions.The most important one of them is a necessary and sufficient condition for the given vector s to be a direction of recession.
引文
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