Adaptive Algorithms for Relatively Lipschitz Continuous Convex Optimization Problems

TL;DR

This paper introduces adaptive algorithms for convex optimization problems with relative Lipschitz continuity, expanding the applicability of gradient methods and providing optimal convergence rates for a broader class of problems.

Contribution

It proposes new adaptive and universal methods for convex minimization under relative Lipschitz conditions, with theoretical convergence guarantees and numerical validation.

Findings

Adaptive methods achieve optimal convergence rates.

Universal method applies to both relatively smooth and Lipschitz continuous functions.

Numerical experiments confirm theoretical results.

Abstract

Recently there were proposed some innovative convex optimization concepts, namely, relative smoothness [1] and relative strong convexity [2,3]. These approaches have significantly expanded the class of applicability of gradient-type methods with optimal estimates of the convergence rate, which are invariant regardless of the dimensionality of the problem. Later Yu. Nesterov and H. Lu introduced some modifications of the Mirror Descent method for convex minimization problems with the corresponding analogue of the Lipschitz condition (so-called relative Lipschitz continuity). By introducing an artificial inaccuracy to the optimization model, we propose adaptive methods for minimizing a convex Lipschitz continuous function, as well as for the corresponding class of variational inequalities. We also consider an adaptive "universal" method, applicable to convex minimization problems both on the class of relatively smooth and relatively Lipschitz continuous functionals with optimal estimates of the convergence rate. The universality of the method makes it possible to justify the applicability of the obtained theoretical results to a wider class of convex optimization problems. We also present the results of numerical experiments.