Linear convergence of epsilon-subgradient descent methods for a class of convex functions

A linear convergence rate is established for a class of epsilon-subgradient descent methods for minimizing certain convex functions on ℝn whose conjugates have subdifferentials that are locally upper Lipschitzian at the origin, a property generalizing classical regularity conditions.

Abstract

This paper establishes a linear convergence rate for a class of epsilon-subgradient descent methods for minimizing certain convex functions on ℝ n . Currently prominent methods belonging to this class include the resolvent (proximal point) method and the bundle method in proximal form (considered as a sequence of serious steps). Other methods, such as a variant of the proximal point method given by Correa and Lemaréchal, can also fit within this framework, depending on how they are implemented. The convex functions covered by the analysis are those whose conjugates have subdifferentials that are locally upper Lipschitzian at the origin, a property generalizing classical regularity conditions.