A cutting method and construction of mixed minimization algorithms on its basis

We propose a cutting method for conditional minimization of convex functions. We show that it is possible to construct mixed minimization algorithms without losing their convergence on the basis of this method with the assistance of other convex programming methods. The qualitative evaluation of an approximation set, involved in this method, allows us to periodically update the approximation sets in the proposed method and mixed algorithms with the purpose of simplifying the problems of construction of iteration points.