Minimization For Vector-Valued Functions In Partial Ordered Space

Optimization has been widely applied in the real world, and the existence of Pareto elements is very important in studying optimization.This article discusses the existence problem of Pareto elements for cone lower semi-continuous from above functions in partial ordered space.We obtain two results on the existence of minimal elements and smallest elements for such functions, where cone-lower semicontinuity is replaced by cone lower semi-continuity from above. We also point out a cone-semicompact set in H.W.Corley’s paper is not necessarily closed.In the first chapter, we first review the research background and recent development of Minimax theory, then, we introduce the history and development of the minimal points in partial ordered space, the related theory of lower semi-continuous from above for scalar functions and cone-lower semi-continuous from above for vector-valued functions.In the second chapter, we discuss the existence problem of minimal points for cone lower semi-continuous from above functions. When the cone is convex and acute, we obtain the existence of minimal points for the cone lower semi-continuous functions under the compact condition.We give an example to show that a cone-semicompact set in H.W.Corley’s paper is not necessarily closed.In the third chapter, we discuss the existence problem of smallest element for cone lower semi-continuous from above functions. When the cone is closed, convex and pointed, we prove the existence of smallest element under the existence of minimal points condition in chapter 2 for properly quasi C-convex functions. Also, we prove that a cone convex function with totally ordered range space has the smallest element.