On the Properties of the Method of Minimization for Convex Functions with Relaxation on the Distance to Extremum

We present a subgradient method of minimization, similar to the method of minimal iterations for solving systems of equations, which inherits from the latter convergence properties on quadratic functions. The proposed algorithm, for a certain set of parameters, coincides with the previously known method of minimizing piecewise linear functions and is an element of the family of minimization methods with relaxation of the distance to extremum, developed by B.T. Polyak, where the step length is calculated based on the predefined minimum value of the function. We link parameters of this method to the constraint on the degree of homogeneity of the function and obtain estimates on its convergence rate on convex functions. We prove that on some classes of functions it converges at the rate of a geometric progression. We also discuss the computational capabilities of this approach for solving problems with high dimension.