Certain Theoretical Researches And Applications Of Recession Cones And Recession Functions

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.