## On the Rate of Local Convergence of High-Order-Infeasible-Path-Following Algorithms for P*-Linear Complementarity Problems (1997)

Venue: | Computational Optimization and Applications |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Zhao97onthe,

author = {Gongyun Zhao and Jie Sun},

title = {On the Rate of Local Convergence of High-Order-Infeasible-Path-Following Algorithms for P*-Linear Complementarity Problems},

journal = {Computational Optimization and Applications},

year = {1997},

volume = {14},

pages = {293--307}

}

### OpenURL

### Abstract

A simple and unified analysis is provided on the rate of local convergence for a class of high-order-infeasible-path-following algorithms for the P -linear complementarity problem (P -LCP). It is shown that the rate of local convergence of a -order algorithm with a centering step is + 1 if there is a strictly complementary solution and ( + 1)=2 otherwise. For the -order algorithm without the centering step the corresponding rates are and =2, respectively. The algorithm without a centering step does not follow the fixed traditional central path. Instead, at each iteration, it follows a new analytic path connecting the current iterate with an optimal solution to generate the next iterate. An advantage of this algorithm is that it does not restrict iterates in a sequence of contracting neighborhoods of the central path.

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Citation Context ...: jsun@nus.edu.sg 1 1. Introduction The local convergence of interior point methods for the monotone linear complementarity problem (MLCP) has been studied, to name just a few, by Bonnans and Gonzaga =-=[2]-=-, McShane [6], Wright and Zhang [16], and Ye and Anstreicher [17] under the assumption that there exists a strictly complementary solution, and by Mizuno [7], Monteiro and Tsuchiya [10], Potra and She... |

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Citation Context ...ocal convergence, we assume thatsis sufficiently small, i.e.,s2 (0;s] wheresis defined in Assumption 1.1 (i). The analysis on global convergence of high-order algorithms can be found, for example, in =-=[4,9,18]-=-. Lemma 2.1: Under Assumption 1.1(i), there exists a constant C 2 such that an iteration of Algorithm 2.1 can reducesto + with +sC 2s+1 : 5 Proof: Using (2.2) we have X(ff)y(ff) = X k=0 k X i=0 X (i) ... |

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Citation Context ...ocal convergence, we assume thatsis sufficiently small, i.e.,s2 (0;s] wheresis defined in Assumption 1.1 (i). The analysis on global convergence of high-order algorithms can be found, for example, in =-=[4,9,18]-=-. Lemma 2.1: Under Assumption 1.1(i), there exists a constant C 2 such that an iteration of Algorithm 2.1 can reducesto + with +sC 2s+1 : 5 Proof: Using (2.2) we have X(ff)y(ff) = X k=0 k X i=0 X (i) ... |

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Citation Context ...bove result. Therefore Assumption 1.1 is valid for the Ps-LCP. We also note that the definition of F() facilitates the so-called infeasible path-following algorithms. See, for instance, Mizuno et al. =-=[8]-=- for details. After we finished the first draft of this paper, Professor Stoer brought to our attention the paper of Stoer, Wechs and Mizuno [13] in which similar results concerning the algorithms wit... |

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Citation Context ... The local convergence of interior point methods for the monotone linear complementarity problem (MLCP) has been studied, to name just a few, by Bonnans and Gonzaga [2], McShane [6], Wright and Zhang =-=[16]-=-, and Ye and Anstreicher [17] under the assumption that there exists a strictly complementary solution, and by Mizuno [7], Monteiro and Tsuchiya [10], Potra and Sheng [11], and Sturm [14] without such... |

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Citation Context ... involving the existence of first and second order derivatives ats= 0 or ae = 0 were also obtained earlier by Monteiro and Tsuchiya [10], Mizuno [7], and Sturm [14] in various situations. It is shown =-=[3,15]-=- that the SHLCP is equivalent to the so-called Ps-linear complementarity problem (Ps-LCP); which includes convex quadratic programs, the MLCP, and some other linear complementarity problems (cf. [1]) ... |

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Citation Context ...ocal convergence, we assume thatsis sufficiently small, i.e.,s2 (0;s] wheresis defined in Assumption 1.1 (i). The analysis on global convergence of high-order algorithms can be found, for example, in =-=[4,9,18]-=-. Lemma 2.1: Under Assumption 1.1(i), there exists a constant C 2 such that an iteration of Algorithm 2.1 can reducesto + with +sC 2s+1 : 5 Proof: Using (2.2) we have X(ff)y(ff) = X k=0 k X i=0 X (i) ... |

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Citation Context ... involving the existence of first and second order derivatives ats= 0 or ae = 0 were also obtained earlier by Monteiro and Tsuchiya [10], Mizuno [7], and Sturm [14] in various situations. It is shown =-=[3,15]-=- that the SHLCP is equivalent to the so-called Ps-linear complementarity problem (Ps-LCP); which includes convex quadratic programs, the MLCP, and some other linear complementarity problems (cf. [1]) ... |

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Citation Context ...-following algorithms. See, for instance, Mizuno et al. [8] for details. After we finished the first draft of this paper, Professor Stoer brought to our attention the paper of Stoer, Wechs and Mizuno =-=[13]-=- in which similar results concerning the algorithms with a centering step were obtained. Since we offer a different (and possibly 3 simpler) proof for those results, it is our belief that our results ... |

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Citation Context ...u.sg 1 1. Introduction The local convergence of interior point methods for the monotone linear complementarity problem (MLCP) has been studied, to name just a few, by Bonnans and Gonzaga [2], McShane =-=[6]-=-, Wright and Zhang [16], and Ye and Anstreicher [17] under the assumption that there exists a strictly complementary solution, and by Mizuno [7], Monteiro and Tsuchiya [10], Potra and Sheng [11], and ... |

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Citation Context ...ht and Zhang [16], and Ye and Anstreicher [17] under the assumption that there exists a strictly complementary solution, and by Mizuno [7], Monteiro and Tsuchiya [10], Potra and Sheng [11], and Sturm =-=[14]-=- without such an assumption. These studies are generally concerned with path-following algorithms and the results state that local quadratic rate of convergence can be achieved for MLCPs with strictly... |

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Citation Context ...s no confusion. The fact that the central path exists and is smooth to arbitrary order fors? 0 for a wide class of (GCP) has been shown in the monograph of Kojima et al [5]. Recently, Stoer and Wechs =-=[12]-=- have analyzed the analyticity of the central path ats= 0 and ae = 0 for the sufficient horizontal linear complementarity problem (SHLCP). Their result indicates that Assumption 1.1 holds for the SHLC... |

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Citation Context ...3,15] that the SHLCP is equivalent to the so-called Ps-linear complementarity problem (Ps-LCP); which includes convex quadratic programs, the MLCP, and some other linear complementarity problems (cf. =-=[1]-=-) as its special cases. The results of Stoer and Wechs [12] indicate that for the Ps-LCP the function z(; g) defined by (1.1) with F() := f(x; y)jP x \Gamma y = q + q; (x; y)s0g (1:3) is an analytic f... |