Minimization method with approximation of constraint zone and epigraph of objective function

We propose a method for solving a convex programming problem which belongs to a class of cutting-plane methods. For construction of sequences this method uses approximation of the feasible set and the epigraph of the objective function of the solved problem. In the method cutting planes are constructed by subgradients of the objective function and functions which define the constrained set. In this case each iteration point can be found by solving a linear programming problem. The proposed method differs from other famous cutting-plane methods by possibility of updating approximation sets due to dropping accumulated constraints. We substantiate the convergence of the method and discuss its numerical realizations.