An Accelerated Smoothing Newton Method with Cubic Convergence for Weighted Complementarity Problems

Smoothing Newton methods, which usually inherit local quadratic convergence rate, have been successfully applied to solve various mathematical programming problems. In this paper, we propose an accelerated smoothing Newton method (ASNM) for solving the weighted complementarity problem (wCP) by refor...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 196; no. 2; pp. 641 - 665
Main Authors: Tang, Jingyong, Zhou, Jinchuan, Zhang, Hongchao
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2023
Springer Nature B.V
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Computational efficiency
Computing costs
Convergence
Engineering
Linear systems
Mathematical programming
Mathematics
Mathematics and Statistics
Newton methods
Nonlinear equations
Nonlinear systems
Operations Research/Decision Theory
Optimization
Smoothing
Theory of Computation
Accelerated smoothing Newton method
Cubic convergence
Weighted complementarity problem
Nonlinear programming
Nonmonotone line search
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:Smoothing Newton methods, which usually inherit local quadratic convergence rate, have been successfully applied to solve various mathematical programming problems. In this paper, we propose an accelerated smoothing Newton method (ASNM) for solving the weighted complementarity problem (wCP) by reformulating it as a system of nonlinear equations using a smoothing function. In spirit, when the iterates are close to the solution set of the nonlinear system, an additional approximate Newton step is computed by solving one of two possible linear systems formed by using previously calculated Jacobian information. When a Lipschitz continuous condition holds on the gradient of the smoothing function at two checking points, this additional approximate Newton step can be obtained with a much reduced computational cost. Hence, ASNM enjoys local cubic convergence rate but with computational cost only comparable to standard Newton’s method at most iterations. Furthermore, a second-order nonmonotone line search is designed in ASNM to ensure global convergence. Our numerical experiments verify the local cubic convergence rate of ASNM and show that the acceleration techniques employed in ASNM can significantly improve the computational efficiency compared with some well-known benchmark smoothing Newton method.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-022-02152-6