Methods for solving systems of linear and convex inequalities based on the Fejér principle

We consider the technique of constructing Fejér contraction mappings used in iterative processes of solving linear and convex systems of inequalities as well as accompanying optimization problems. The general approach is based on the notion of $M$-Fejér step "$p\to q$" defined by the property
$$ |q-y|<|p-y|,\qquad\forall y\in M. $$
This property (postulate) assumes that $p\not\in\overline{\operatorname{conv}M}$ with sufficiently arbitrary $q\not=\varnothing$. Some of the problems considered in the paper are illustrated by schemes reflecting the analytics of these problems.