## Local Convergence of Predictor-Corrector Infeasible-Interior-Point Algorithms for SDPs and SDLCPs (1997)

Venue: | Mathematical Programming |

Citations: | 53 - 3 self |

### BibTeX

@INPROCEEDINGS{Kojima97localconvergence,

author = {Masakazu Kojima and Masayuki Shida and Susumu Shindoh},

title = {Local Convergence of Predictor-Corrector Infeasible-Interior-Point Algorithms for SDPs and SDLCPs},

booktitle = {Mathematical Programming},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

. An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno-Todd-Ye type predictorcorrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno-Todd-Ye type predictor-corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. Key words. Semidefinite Programming, Infeasible-Interior-Point Method, Predictor-CorrectorMethod, Superlinear Convergence, Primal-Dual Nondegeneracy Abbreviated Title. Interior-Point Algorithms for SDPs y Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japa...

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Citation Context ...;Y ) is an optimal solution of the SDP. ffl Strong Duality: If F " (S ++ 2 S++ ) 6= ; then there exists an (X;Y ) satisfying (2); hence (X;Y ) gives an optimal solution of the SDP. (See, for exam=-=ple, [1]-=-). In general, given an n(n + 1)=2-dimensional affine subspace F of S 2 S, we call the problem of finding an (X;Y ) 2 S 2 S which satisfies (2) an SDLCP [18]. Throughout the paper we assume that the a... |

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Citation Context ...rgence of a Mizuno-Todd-Ye type predictorcorrector primal-dual infeasible-interior-point algorithm with the use of the search direction proposed originally by Helmberg, Rendl, Vanderbei and Wolkowicz =-=[9]-=- for the SDP and independently by Kojima, Shindoh and Hara [18] for the monotone SDLCP (semidefinite linear complementarity problem). So far, many interior-point algorithms have been developed for the... |

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Citation Context ... (C). It is interesting to investigate whether some other predictor directions attain superlinear convergence without condition (C). In a numerical experiment on the Nesterov-Todd predictor direction =-=[28, 29] applied-=- to Example 3.3, we observed that the step length �� ff r to the boundary of the feasible region did not converge to 1:0 but to some positive number �� ff less than 1:0; hence it is unlikely t... |

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Citation Context ... infeasible-interior-point algorithm with the use of the search direction proposed originally by Helmberg, Rendl, Vanderbei and Wolkowicz [9] for the SDP and independently by Kojima, Shindoh and Hara =-=[18]-=- for the monotone SDLCP (semidefinite linear complementarity problem). So far, many interior-point algorithms have been developed for the SDP ([1, 2, 6, 9, 10, 27, 33, 34, etc.]). But no local converg... |

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Citation Context ...thm (Algorithm I) given by Mizuno-Kojima-Todd [22] for the LP to the monotone SDLCP (2). Quite recently, much progress has been made in the primal-dual interior-point algorithms for the SDP. Monteiro =-=[24]-=- devised a new formulation of the primal-dual search direction introduced originally in [9] and studied independently in [18]. He amended key inequalities used to evaluate the computational complexity... |

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Citation Context ... we discuss in Section 5, to ensure superlinear convergence. Our nondegeneracy condition is closely related to the primal and the dual nondegeneracy conditions given by Alizadeh, Haeberly and Overton =-=[3]-=-. Section 6 is devoted to a proof of the superlinear convergence of the algorithm given in Section 4. Remark 1.1. The papers [24, 31, 37] dealt with the following primal-dual pair of SDPs:sP : minimiz... |

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Citation Context ...ree types of interior-point algorithms for monotone SDLCPs. (a) A central trajectory following feasible-interior-point algorithm which is an extension of the algorithms given by Kojima-Mizuno-Yoshise =-=[14]-=- for the monotone LCP (linear complementarity problem) and Monteiro-Adler [25] for the LP (linear program) to the monotone SDLCP (2). (b) A potential-reduction feasible-interior-point algorithm which ... |

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Citation Context ...iro-Adler [25] for the LP (linear program) to the monotone SDLCP (2). (b) A potential-reduction feasible-interior-point algorithm which is an extension of the algorithm given by Kojima-Mizuno-Yoshise =-=[15]-=- for the monotone LCP to the monotone SDLCP (2). (c) A potential-reduction infeasible-interior-point algorithm which is an extension of the constrained potential reduction algorithm (Algorithm I) give... |

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Citation Context ...oint algorithm which is an extension of the algorithm given by Zhang [36] for the monotone horizontal LCP to the SDP. Using Monteiro's formulation of the primal-dual search direction, Potra and Sheng =-=[31]-=- proposed (f) a globally convergent predictor-corrector primal-dual infeasible-interior-point algorithm which is an extension of the Mizuno-Todd-Ye type predictor-corrector interior-point algorithm [2... |

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Citation Context ...es. Potra and Sheng [32] proved the superlinear convergence of their predictor-corrector algorithm (f) to which we have referred in the Introduction under conditions (A) and (C). Luo, Sturm and Zhang =-=[20]-=- proved superlinear convergence for the Nesterov-Todd search direction [28, 29] under conditions (A) and (C). More recently, in a revised version of their paper [17] Kojima, Shida and Shindoh proved t... |

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Citation Context ...the LP to the SDP. Zhang [37] polished and deepened Monteiro's analysis further. He presented (e) a long-step infeasible-interior-point algorithm which is an extension of the algorithm given by Zhang =-=[36]-=- for the monotone horizontal LCP to the SDP. Using Monteiro's formulation of the primal-dual search direction, Potra and Sheng [31] proposed (f) a globally convergent predictor-corrector primal-dual i... |

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Citation Context ...n, and presented (d) a long-step central trajectory following feasible-interior-point algorithm which is an extension of the algorithm given by Kojima-Mizuno-Yoshise [13] for the LP to the SDP. Zhang =-=[37]-=- polished and deepened Monteiro's analysis further. He presented (e) a long-step infeasible-interior-point algorithm which is an extension of the algorithm given by Zhang [36] for the monotone horizon... |

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Citation Context ...ntially to the central surface. See (23). In the LP case, we do not need any of these additional conditions to attain superlinear convergence [35, etc.]. In the monotone LCP case, Monteiro and Wright =-=[26]-=- showed that the standard predictor-corrector interior-point algorithm can not enjoy superlinear convergence without the strict complementarity condition (A). Mizuno [21] proposed a modification of th... |

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Citation Context ... (A) and (C). Luo, Sturm and Zhang [20] proved superlinear convergence for the Nesterov-Todd search direction [28, 29] under conditions (A) and (C). More recently, in a revised version of their paper =-=[17]-=- Kojima, Shida and Shindoh proved the quadratic convergence of a predictor-corrector algorithm using the Alizadeh-Haeberly-Overton search direction [2, 4] under condition (A). Acknowledgment: The auth... |

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Citation Context ... (C). It is interesting to investigate whether some other predictor directions attain superlinear convergence without condition (C). In a numerical experiment on the Nesterov-Todd predictor direction =-=[28, 29] applied-=- to Example 3.3, we observed that the step length �� ff r to the boundary of the feasible region did not converge to 1:0 but to some positive number �� ff less than 1:0; hence it is unlikely t... |

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Citation Context ... the monotone SDLCP (2). (c) A potential-reduction infeasible-interior-point algorithm which is an extension of the constrained potential reduction algorithm (Algorithm I) given by Mizuno-Kojima-Todd =-=[22]-=- for the LP to the monotone SDLCP (2). Quite recently, much progress has been made in the primal-dual interior-point algorithms for the SDP. Monteiro [24] devised a new formulation of the primal-dual ... |

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Citation Context ...LCP case, Monteiro and Wright [26] showed that the standard predictor-corrector interior-point algorithm can not enjoy superlinear convergence without the strict complementarity condition (A). Mizuno =-=[21]-=- proposed a modification of the predictorcorrector interior-point algorithm that ensures superlinear convergence without the strict complementarity condition (A). In the SDP case, the strict complemen... |

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Citation Context ... Nesterov-Todd direction would attain superlinear convergence without condition (C). On the other hand, the nondegeneracy condition (B) turned out to be unnecessary in recent studies. Potra and Sheng =-=[32]-=- proved the superlinear convergence of their predictor-corrector algorithm (f) to which we have referred in the Introduction under conditions (A) and (C). Luo, Sturm and Zhang [20] proved superlinear ... |

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Citation Context ...dictor direction 0 for the corrector direction (i = 1; 2; : : : ; m); (x 0 ; X 0 ; Y 0 ) : the initial point: Because of this advantage in computation, the system (11) was used in the implementations =-=[7, 11]-=- of the interior-point algorithm described in [18]. If we employ the new representation of the primal-dual search direction recently given by Monteiro [24], we can eliminate the skew-symmetric variabl... |

2 |
Private communication
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Citation Context ...lementarity condition (ii), the nondegeneracy condition (iii) is equivalent to the combination of the primal and the dual nondegeneracy conditions given in the paper [3]. This fact is due to Haeberly =-=[8]-=-. Remark. We can represent each W 2 S as W = W + +W 0 ; W + 2 S+ ; 0W 0 2 S+ and W + ffl (0W 0 ) = 0: Here W + denotes the orthogonal projection of W 2 S onto the positive semidefinite cone S+ and W 0... |