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arXiv · 2009.10551

A Generalized Newton Method for Subgradient Systems

Pham Duy Khanh·Boris Mordukhovich·Vo Thanh Phat
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Abstract

This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order subdifferential of such functions that enjoys extensive calculus rules and can be efficiently computed for broad classes of extended-real-valued functions. Based on this and on metric regularity and subregularity properties of subgradient mappings, we establish verifiable conditions ensuring well-posedness of the proposed algorithm and its local superlinear convergence. The obtained results are also new for the class of equations defined by continuously differentiable functions with Lipschitzian gradients (${\cal C}^{1,1}$ functions), which is the underlying case of our consideration. The developed algorithms for prox-regular functions and its extension to a structured class of composite functions are formulated in terms of proximal mappings and forward-backward envelopes. Besides numerous illustrative examples and comparison with known algorithms for ${\cal C}^{1,1}$ functions and generalized equations, the paper presents applications of the proposed algorithms to regularized least square problems arising in statistics, machine learning, and related disciplines.

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Pham Duy Khanh, Boris Mordukhovich, Vo Thanh Phat. 2020-09-22. A Generalized Newton Method for Subgradient Systems. https://arxiv.org/abs/2009.10551

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