A Regularized Semi-Smooth Newton Method with Projection Steps for Composite Convex Programs

The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: (1) Many well-known operator splitting methods, such as forward–backward splitting and Douglas–Rachford splitting, actually defin...

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Vydáno v:	Journal of scientific computing Ročník 76; číslo 1; s. 364 - 389
Hlavní autoři:	Xiao, Xiantao, Li, Yongfeng, Wen, Zaiwen, Zhang, Liwei
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.07.2018 Springer Nature B.V
Témata:	Algorithms Computational Mathematics and Numerical Analysis Convergence Convex analysis Hyperplanes Jacobi matrix method Jacobian matrix Machine learning Mapping Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Newton methods Nonlinear equations Optimization Partial differential equations Regularization Splitting Theoretical Semi-smoothness Operator splitting methods 90C30 65K05 Proximal mapping Newton method Composite convex programs
ISSN:	0885-7474, 1573-7691
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Shrnutí:	The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: (1) Many well-known operator splitting methods, such as forward–backward splitting and Douglas–Rachford splitting, actually define a fixed-point mapping; (2) The optimal solutions of the composite convex program and the solutions of a system of nonlinear equations derived from the fixed-point mapping are equivalent. Solving this kind of system of nonlinear equations enables us to develop second-order type methods. These nonlinear equations may be non-differentiable, but they are often semi-smooth and their generalized Jacobian matrix is positive semidefinite due to monotonicity. By combining with a regularization approach and a known hyperplane projection technique, we propose an adaptive semi-smooth Newton method and establish its convergence to global optimality. Preliminary numerical results on ℓ 1 -minimization problems demonstrate that our second-order type algorithms are able to achieve superlinear or quadratic convergence.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0885-7474 1573-7691
DOI:	10.1007/s10915-017-0624-3