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 Sub-quadratic convergence of a smoothing Newton algorithm for the P(0) and monotone LCP Huang, ZH; Qi, LQ; Sun, DF 2004-04-01 Source Publication MATHEMATICAL PROGRAMMING ISSN 0025-5610 Volume 99Issue:3Pages:423-441 Abstract Given M is an element of R(nxn) and q is an element of R(n), the linear complementarity problem (LCP) is to find (x, s) is an element of R(n) x R(n) such that (x, s) greater than or equal to 0, s = M(x) + q, x(T)s = 0. By using the Chen-Harker-Kanzow-Sniale (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 [Matheniatical 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 is an element of (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. Keyword linear complementarity problem smoothing Newton method global convergence sub-quadratic convergence DOI 10.1007/s10107-003-0457-8 Language 英语 WOS Research Area Computer Science ; Operations Research & Management Science ; Mathematics WOS Subject Computer Science, Software Engineering ; Operations Research & Management Science ; Mathematics, Applied WOS ID WOS:000220991700002 Publisher SPRINGER Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/714 Collection 中国科学院数学与系统科学研究院 Corresponding Author Huang, ZH Affiliation 1.Chinese Acad Sci, Acad Math & Syst Sci, Inst Appl Math, Beijing 100080, Peoples R China2.Hong Kong Polytech Univ, Dept Appl Math, Hong Kong, Hong Kong, Peoples R China3.Natl Univ Singapore, Dept Math, Singapore 117543, Singapore Recommended CitationGB/T 7714 Huang, ZH,Qi, LQ,Sun, DF. Sub-quadratic convergence of a smoothing Newton algorithm for the P(0) and monotone LCP[J]. MATHEMATICAL PROGRAMMING,2004,99(3):423-441. APA Huang, ZH,Qi, LQ,&Sun, DF.(2004).Sub-quadratic convergence of a smoothing Newton algorithm for the P(0) and monotone LCP.MATHEMATICAL PROGRAMMING,99(3),423-441. MLA Huang, ZH,et al."Sub-quadratic convergence of a smoothing Newton algorithm for the P(0) and monotone LCP".MATHEMATICAL PROGRAMMING 99.3(2004):423-441.
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