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Sub-quadratic convergence of a smoothing Newton algorithm for the P 0 – and monotone LCP

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Abstract.

Given , the linear complementarity problem (LCP) is to find such that (x, s)≥ 0,s=Mx+q,x Ts=0. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the LCP is reformulated as a system of parameterized smooth-nonsmooth equations. As a result, a smoothing Newton algorithm, which is a modified version of the Qi-Sun-Zhou algorithm [Mathematical Programming, Vol. 87, 2000, pp. 1–35], is proposed to solve the LCP with M being assumed to be a P 0 -matrix (P 0 –LCP). The proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration. It is proved in this paper that the proposed algorithm has the following convergence properties: (i) it is well-defined and any accumulation point of the iteration sequence is a solution of the P 0 –LCP; (ii) it generates a bounded sequence if the P 0 –LCP has a nonempty and bounded solution set; (iii) if an accumulation point of the iteration sequence satisfies a nonsingularity condition, which implies the P 0 –LCP has a unique solution, then the whole iteration sequence converges to this accumulation point sub-quadratically with a Q-rate 2–t, where t∈(0,1) is a parameter; and (iv) if M is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition, then the whole sequence converges to the accumulation point quadratically.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, P.R. China

    Zheng-Hai Huang

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

    Liqun Qi

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Defeng Sun

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  1. Zheng-Hai Huang
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  2. Liqun Qi
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  3. Defeng Sun
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Correspondence to Defeng Sun.

Additional information

This author’s work is supported by the Hong Kong Research Grant Council and the Australian Research Council.

This author’s work is supported by Grant R146-000-035-101 of National University of Singapore.

Mathematics Subject Classification (1991): 90C33, 65K10

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Huang, ZH., Qi, L. & Sun, D. Sub-quadratic convergence of a smoothing Newton algorithm for the P 0 – and monotone LCP. Math. Program., Ser. A 99, 423–441 (2004). https://doi.org/10.1007/s10107-003-0457-8

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