On Some Methods for Strongly Convex Optimization Problems with One Functional Constraint

摘要

We consider the classical optimization problem of minimizing a strongly convex, non-smooth, Lipschitz-continuous function with one Lipschitz-continuous constraint. We develop the approach in [10] and propose two methods for the considered problem with adaptive stopping rules. The main idea of the methods is using the dichotomy method and solving an auxiliary one-dimensional problem at each iteration. Theoretical estimates for the proposed methods are obtained. Partially, for smooth functions, we prove the linear rate of convergence of the methods. We also consider theoretical estimates in the case of non-smooth functions. The results for some examples of numerical experiments illustrating the advantages of the proposed methods and the comparison with some adaptive optimal method for non-smooth strongly convex functions are also given.