The linear convergence of a derivative-free descent method for nonlinear complementarity problems

Wei-Zhe Gu; Li-Yong Lu

doi:10.3934/jimo.2016030

Article Contents

2017, Volume 13, Issue 2: 531-548. Doi: 10.3934/jimo.2016030

This issue Previous Article Next Article On fractional vector optimization over cones with support functions

The linear convergence of a derivative-free descent method for nonlinear complementarity problems

Wei-Zhe Gu^1, and
Li-Yong Lu^2, ,

1.
Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
2.
Department of Mathematics, School of Science, Tianjin University of Technology, Tianjin 300384, China

* Corresponding author: Li-Yong Lu
* Corresponding author: Li-Yong Lu

Received: October 31, 2012

Revised: February 29, 2016

Published: August 2017

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Abstract

Recently, Hu, Huang and Chen [Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math. 230 (2009): 69-82] introduced a family of generalized NCP-functions, which include many existing NCP-functions as special cases. They obtained several favorite properties of the functions; and by which, they showed that a derivative-free descent method is globally convergent under suitable assumptions. However, no result on convergent rate of the method was reported. In this paper, we further investigate some properties of this family of generalized NCP-functions. In particular, we show that, under suitable assumptions, the iterative sequence generated by the descent method discussed in their paper converges globally at a linear rate to a solution of the nonlinear complementarity problem. Some preliminary numerical results are reported, which verify the theoretical results obtained.

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Mathematics Subject Classification: Primary: 90C33; Secondary: 65K10.

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Figure 1. Convergence behavior of "gafni(1)" with p = 1.1

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Figure 2. Convergence behavior of "gafni(1)" with p = 10

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Figure 3. Convergence behavior of "gafni(1)" with p = 100

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Figure 4. Convergence behavior of "josephy(5)" with θ = 0

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Figure 5. Convergence behavior of "josephy(5)" with θ = 0.5

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Figure 6. Convergence behavior of "josephy(5)" with θ = 1

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