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 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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01.07.2018
Springer Nature B.V |
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| ISSN: | 0885-7474, 1573-7691 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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 ℓ1-minimization problems demonstrate that our second-order type algorithms are able to achieve superlinear or quadratic convergence. 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. |
| Author | Zhang, Liwei Xiao, Xiantao Wen, Zaiwen Li, Yongfeng |
| Author_xml | – sequence: 1 givenname: Xiantao surname: Xiao fullname: Xiao, Xiantao organization: School of Mathematical Sciences, Dalian University of Technology – sequence: 2 givenname: Yongfeng surname: Li fullname: Li, Yongfeng organization: School of Mathematical Sciences, Peking University – sequence: 3 givenname: Zaiwen orcidid: 0000-0003-1762-0671 surname: Wen fullname: Wen, Zaiwen email: wenzw@pku.edu.cn organization: Beijing International Center for Mathematical Research, Peking University – sequence: 4 givenname: Liwei surname: Zhang fullname: Zhang, Liwei organization: School of Mathematical Sciences, Dalian University of Technology |
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| Keywords | Semi-smoothness Operator splitting methods 90C30 65K05 Proximal mapping Newton method Composite convex programs |
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| SubjectTerms | 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 |
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| Title | A Regularized Semi-Smooth Newton Method with Projection Steps for Composite Convex Programs |
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