A Two-Step Newton Algorithm for the Weighted Complementarity Problem with Local Biquadratic Convergence

We discuss the weighted complementarity problem, extending the nonlinear complementarity problem on Rn. In contrast to the NCP, many equilibrium problems in science, engineering, and economics can be transformed into WCPs for more efficient methods. Smoothing Newton algorithms, known for their at le...

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Bibliographic Details
Published in:Axioms Vol. 12; no. 9; p. 897
Main Authors: Liu, Xiangjing, Liu, Yihan, Zhang, Jianke
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.09.2023
Subjects:
Algorithms
Convergence
Convergence (Mathematics)
derivative-free line search
Iterative methods
Linear programming
local biquadratic convergence
Market equilibrium
Nonlinear theories
two-step Newton method
weighted complementarity problem
China
ISSN:2075-1680, 2075-1680
Online Access:Get full text
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Summary:We discuss the weighted complementarity problem, extending the nonlinear complementarity problem on Rn. In contrast to the NCP, many equilibrium problems in science, engineering, and economics can be transformed into WCPs for more efficient methods. Smoothing Newton algorithms, known for their at least locally superlinear convergence properties, have been widely applied to solve WCPs. We suggest a two-step Newton approach with a local biquadratic order convergence rate to solve the WCP. The new method needs to calculate two Newton equations at each iteration. We also insert a new term, which is of crucial importance for the local biquadratic convergence properties when solving the Newton equation. We demonstrate that the solution to the WCP is the accumulation point of the iterative sequence produced by the approach. We further demonstrate that the algorithm possesses local biquadratic convergence properties. Numerical results indicate the method to be practical and efficient.
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ISSN:2075-1680
2075-1680
DOI:10.3390/axioms12090897