Variational Analysis and Set Optimization

Chapter 11

Abstract Convexity and Solvability Theorems

Ali Reza Doagooei

Department of Applied Mathematics, Shahid Bahonar University of Kerman, Kerman Iran. [e-mail]

11.1  Introduction

One of the main results from convex analysis states that every convex, proper and lower semi-continuous function is the upper envelope of a set of affine functions, that is

$f\left(x\right)=\sup \left\{h\left(x\right):h\text{is an affine function and }h\le f\right\}.$

(11.1)

This is concluded from the fact that a point out of a closed convex set can be separated from that set by an affine function (geometrically by a hyperplane). Since affine functions are constructed by linear functions, the aforementioned results demonstrate that linear functions are instrumental for studying convexity. ...