Subgradient and ε-Subgradient Methods

Abstract

Let us consider a convex programming problem (CPP):

$$find{f^*} = \inf {f_0}\left( x \right),x = \left( {{x^{\left( 1 \right)}},...,{x^{\left( n \right)}}} \right) \in {E^n},$$

(2.1)

subject to constraints:

$${f_i}\left( x \right)\quad 0,\quad i \in \left\{ {1,2, \ldots ,m} \right\} = I;$$

(2.2)

$$x \in X\quad \subseteq {E^n},$$

(2.3)

where F _ν (x), ν ∈ {{0} ∪ I}, are proper convex fuctions determined on convex open neigborhood X ¹ of a given convex set X. We introduce the notion of “information portion” for this problem.