A relaxed version of the cutting method with approximation of the constraint region

A cutting method was proposed for solving the convex programming problem. The method assumes that the constraint region of the problem is embedded into some polyhedral sets for constructing iteration points. It involves the construction of a sequence of approximations that belongs to the admissible set and is relaxed, as well as implies that the $\varepsilon$-solution of the initial problem is fixed after a finite number of steps. The method also allows to obtain mixed convergent algorithms by using, if desired, any known or new relaxation algorithms for constructing the main iteration points.