## Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming (1996)

Venue: | Journal of Optimization Theory and Applications |

Citations: | 20 - 7 self |

### BibTeX

@ARTICLE{Potra96superlinearconvergence,

author = {Florian A. Potra and Rongqin Sheng},

title = {Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming},

journal = {Journal of Optimization Theory and Applications},

year = {1996},

volume = {86}

}

### Years of Citing Articles

### OpenURL

### Abstract

We prove the superlinear convergence of the primal-dual infeasible-interior-point path-following algorithm proposed recently by Kojima, Shida and Shindoh and the present authors, under two conditions: (1) the SDP problem has a strictly complementary solution, and (2) the size of the central path neighborhood approaches zero. The nondegeneracy condition suggested by Kojima, Shida and Shindoh is not used in our analysis. Our result implies that the modified algorithm of Kojima, Shida and Shindoh, which enforces condition (2) by using additional corrector steps, has superlinear convergence under the standard assumption of strict complementarity. Finally, we point out that condition (2) can be made weaker and show the superlinear convergence under the strict complementarity assumption and a weaker condition than (2). Key Words: semidefinite programming, path-following, infeasible-interior-point algorithm, polynomiality, superlinear convergence. Abbreviated Title: Superlinear convergence ...

### Citations

499 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
- 1995
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Citation Context ...thematics, University of Iowa, Iowa City, IA 52242, USA. 1 Introduction Many primal-dual interior-point algorithms have been proposed recently for solving semidefinite programming (SDP) problems (cf. =-=[1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15]-=-). Most of these algorithms use one of the following three search directions: the Kojima-Shindoh-Hara direction [6], the Alizadeh-Haeberly-Overton direction [1] and the Nesterov-Todd direction [10]. U... |

215 | An interior-point method for semidefinite programming
- Helmberg, Rendl, et al.
- 1996
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Citation Context ...thematics, University of Iowa, Iowa City, IA 52242, USA. 1 Introduction Many primal-dual interior-point algorithms have been proposed recently for solving semidefinite programming (SDP) problems (cf. =-=[1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15]-=-). Most of these algorithms use one of the following three search directions: the Kojima-Shindoh-Hara direction [6], the Alizadeh-Haeberly-Overton direction [1] and the Nesterov-Todd direction [10]. U... |

197 | A primal-dual interior point algorithm for linear programming
- Kojima, Mizuno, et al.
- 1989
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Citation Context ...rithm In a recent paper [12], we proposed an infeasible-interior-point algorithm for solving (2.3), which generalizes the interior--point method for linear programming proposed by Mizuno, Todd and Ye =-=[8]. The algorithm performs-=- in a neighborhood of the infeasible central path: C(��) = fZ = (X; y; S) 2 S n ++ \Theta IR m \Theta S n ++ : XS = ��I; R i = (��=�� 0 )R 0 i ; i = 1; : : : ; m; R d = (��=�� ... |

189 | Primal-dual interior-point methods for self-scaled cones
- Nesterov, Todd
- 1998
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Citation Context ...thematics, University of Iowa, Iowa City, IA 52242, USA. 1 Introduction Many primal-dual interior-point algorithms have been proposed recently for solving semidefinite programming (SDP) problems (cf. =-=[1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15]-=-). Most of these algorithms use one of the following three search directions: the Kojima-Shindoh-Hara direction [6], the Alizadeh-Haeberly-Overton direction [1] and the Nesterov-Todd direction [10]. U... |

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47 | On extending primal-dual interior-point algorithms from linear programming to semidefinite programming
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35 | On homogeneous interior-point algorithms for semidefinite programming
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29 | A predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem using the Alizadeh–Haeberly–Overton search direction
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- 1994
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11 | Global and local convergence of predictorcorrector infeasible--interior--point algorithms for semidefinite programs
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- 1995
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