A Globally Convergent Proximal Newton-Type Method in Nonsmooth Convex Optimization

TL;DR

This paper introduces a new proximal Newton-type algorithm for nonsmooth convex optimization that guarantees global convergence and superlinear local convergence rates without requiring strong convexity or Lipschitz continuity of Hessians.

Contribution

It develops a novel algorithm with proven global and local convergence properties applicable to a wide class of nonsmooth convex problems, even with structured non-Lipschitz Hessians.

Findings

Algorithm achieves global convergence.

Local convergence is superlinear or quadratic.

Numerical experiments confirm practical effectiveness.

Abstract

The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of variational analysis, we establish implementable results on the global convergence of the proposed algorithm as well as its local convergence with superlinear and quadratic rates. For certain structured problems, the obtained local convergence conditions do not require the local Lipschitz continuity of the corresponding Hessian mappings that is a crucial assumption used in the literature to ensure a superlinear convergence of other algorithms of the proximal Newton type. The conducted numerical experiments of solving the $l_1$ regularized logistic regression model illustrate the possibility of applying the proposed algorithm to deal with practically important problems.