A Relaxed Cutting Method for the Convex Programming Problem

We propose a method for constrained minimization of a convex non-differentiable function, which belongs to the class of cutting methods. To construct iteration points, it uses the immersion operation into polyhedral sets of both the constraint set of the original problem and the epigraph of its objective function. The method is characterized by the fact that the main sequence of iteration points is constructed belonging to an admissible set with a relaxation condition. In this regard, it is permissible to check each iteration point for e-optimality. In addition, the method includes the possibility of combining it with other relaxation algorithms. The convergence of the proposed method is substantiated and its implementations are described.