Globally Convergent Coderivative-Based Generalized Newton Methods in Nonsmooth Optimization

TL;DR

This paper introduces two globally convergent Newton-type algorithms for nonsmooth optimization, utilizing coderivatives and generalized Hessians, with proven convergence rates and numerical validation on Lasso and quadratic programming problems.

Contribution

The paper develops and justifies two new globally convergent Newton-type methods based on coderivatives for nonsmooth optimization, extending existing techniques with superlinear convergence under certain conditions.

Findings

Algorithms demonstrate at least linear convergence.

Superlinear convergence achieved under semismooth* property.

Numerical experiments show competitive performance on benchmark problems.

Abstract

This paper proposes and justifies two globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are coderivative-based and employ generalized Hessians (coderivatives of subgradient mappings) associated with objective functions, which are either of class $\mathcal{C}^{1,1}$, or are represented in the form of convex composite optimization, where one of the terms may be extended-real-valued. The proposed globally convergent algorithms are of two types. The first one extends the damped Newton method and requires positive-definiteness of the generalized Hessians for its well-posedness and efficient performance, while the other algorithm is of {the regularized Newton type} being well-defined when the generalized Hessians are merely positive-semidefinite. The obtained convergence rates for both methods are at least linear, but become superlinear under the semismooth$^*$ property of subgradient mappings. Problems of convex composite optimization are investigated with and without the strong convexity assumption {on smooth parts} of objective functions by implementing the machinery of forward-backward envelopes. Numerical experiments are conducted for Lasso problems and for box constrained quadratic programs with providing performance comparisons of the new algorithms and some other first-order and second-order methods that are highly recognized in nonsmooth optimization.