## Primal-Dual Path-Following Algorithms for Semidefinite Programming (1996)

Venue: | SIAM Journal on Optimization |

Citations: | 151 - 10 self |

### BibTeX

@ARTICLE{Monteiro96primal-dualpath-following,

author = {Renato D.C. Monteiro},

title = {Primal-Dual Path-Following Algorithms for Semidefinite Programming},

journal = {SIAM Journal on Optimization},

year = {1996},

volume = {7},

pages = {663--678}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper deals with a class of primal-dual interior-point algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh and Hara [11]. These authors proposed a family of primal-dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X 1=2 S \Gamma1=2 . They study three primaldual algorithms based on this family of search directions: a short-step path-following method, a feasible potential-reduction method and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno and Yoshise [10]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present: 1) a simplified polynomial convergence proof for one of their short-step path-following ...

### Citations

4984 |
Matrix Analysis
- Horn, Johnson
- 1991
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Citation Context ...ansformed into an algorithm for SDP in a mechanical way. Since then several authors have proposed interior-point algorithms for solving SDP problems including Helmberg, Rendl, Vanderbei and Wolkowicz =-=[5]-=-, Jarre [8], Kojima, Shindoh and Hara [11], Nesterov and Nemirovskii [16], Nesterov and Todd [19, 18] and Vandenberghe and Boyd [20]. Among the above works, Kojima, Shindoh and Hara [11] and Nesterov ... |

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- 1989
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Citation Context ...hod and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno and Yoshise =-=[10]-=-. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present... |

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182 | Interior path following primal-dual algorithms. Part I: Linear programming - Monteiro, Adler - 1989 |

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A Polynomial-time Algorithm for a Class of Linear Complementarity Problems
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Citation Context ...00014-93-1-0234 and N00014-941 -0340. Their financial support is gratefully acknowledged. y School of Industrial and Systems Engineering, Georgia Tech, Atlanta, GA 30332. (monteiro@isye.gatech.edu) 1 =-=[9]-=- and Monteiro and Adler [12, 13], referred in here to as the short-step path-following method, improves the worst-case iteration complexity of the algorithm of [10] by a factor of p n by generating it... |

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A class of projective transformations for linear programming
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Citation Context ...941 -0340. Their financial support is gratefully acknowledged. y School of Industrial and Systems Engineering, Georgia Tech, Atlanta, GA 30332. (monteiro@isye.gatech.edu) 1 [9] and Monteiro and Adler =-=[12, 13]-=-, referred in here to as the short-step path-following method, improves the worst-case iteration complexity of the algorithm of [10] by a factor of p n by generating iterates in a narrower neighborhoo... |

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Citation Context ... path. Several authors have discussed generalizations of interior-point algorithms for linear programming to the context of SDP. The landmark work in this direction is due to Nesterov and Nemirovskii =-=[14, 15]-=- where a general approach for using interior-point methods for solving convex programs is proposed based on the notion of self-concordant functions. (See their book [17] for a comprehensive treatment ... |

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