Nonlinear Scalarization ofε-Properly Efficient Solutions
Xia Yuanme · Chongqing Shifan Daxue xuebao. Ziran kexue ban · 2015
In this paper,we study some nonlinear scalarization characterizations ofe-properly efficient solutions for vector optimization problems via the classical nonlinear scalarization functionΔ-K obtained by using apointed closed convex cone.We first prove thate-properly efficient solutions of the vector optimization problem(VP)implies de+K(0)-approximate solutions of the scalarization problem(Py)and give an example to illustrate the fact that the converse of this conclusion may not be valid.Furthermore,we also prove that strictlyβ-approximate solutions of the scalarization problem(Py)impliese-properly efficient solutions of the vector optimization problem(VP),and propose some examples to illustrate the facts that this conclusion may not be true if the cone hull of the set f(S)+e+K-f(x)is not closed,andβ-approximate solutions of the scalarization problem(Py)does not implye-properly efficient solutions of the vector optimization problem(VP).