# A globally convergent proximal Newton-type method in nonsmooth convex optimization

@article{Mordukhovich2022AGC, title={A globally convergent proximal Newton-type method in nonsmooth convex optimization}, author={Boris S. Mordukhovich and Xiaoming Yuan and Shangzhi Zeng and Jin Zhang}, journal={Mathematical Programming}, year={2022} }

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 structural problems, the obtained local convergence conditions do not…

## 7 Citations

Generalized Damped Newton Algorithms in Nonsmooth Optimization with Applications to Lasso Problems

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New globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems with extended-real-valued cost functions, which typically arise in machine learning and statistics.

ADVANCES IN CONVERGENCE AND SCOPE OF THE PROXIMAL POINT ALGORITHM

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- 2020

The proximal point algorithm, as an approach to finding a zero of a maximal monotone mapping, is well known for its role in numerical optimization, such as in methods of multipliers (ALM). Although…

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- 2022

This paper is concerned with lq (0<q<1)-norm regularized minimization problems with a twice continuously differentiable loss function. For this class of nonconvex and nonsmooth composite problems,…

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A method that makes minimal assumptions about the two functions, does not require the tuning of algorithm parameters, and works well in practice across a variety of problems, showing that the method is more efficient than standard solvers when the oracle function contains much data.

Globally Convergent Coderivative-Based Generalized Newton Methods in Nonsmooth Optimization

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- 2021

This paper proposes and justifies two new globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and…

Generalized Damped Newton Algorithms in Nonsmooth Optimization via Second-Order Subdifferentials

- Mathematics, Computer Science
- 2021

New globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems with extended-real-valued cost functions, which typically arise in machine learning and statistics.

Estimates of Generalized Hessians for Optimal Value Functions in Mathematical Programming

- Mathematics, Computer ScienceSet-Valued and Variational Analysis
- 2021

The main goal of this paper is to provide estimates of the generalized Hessian for the optimal value function, which could enable the development of robust solution algorithms, such as the Newton method.

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