## Homogeneous Interior-Point Algorithms for Semidefinite Programming (1995)

Venue: | Department of Mathematics, The University of Iowa |

Citations: | 33 - 8 self |

### BibTeX

@ARTICLE{Potra95homogeneousinterior-point,

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

title = {Homogeneous Interior-Point Algorithms for Semidefinite Programming},

journal = {Department of Mathematics, The University of Iowa},

year = {1995},

volume = {82},

pages = {161--184}

}

### Years of Citing Articles

### OpenURL

### Abstract

A simple homogeneous primal-dual feasibility model is proposed for semidefinite programming (SDP) problems. Two infeasible-interior-point algorithms are applied to the homogeneous formulation. The algorithms do not need big M initialization. If the original SDP problem has a solution, then both algorithms find an ffl-approximate solution (i.e., a solution with residual error less than or equal to ffl) in at most O( p n ln(ae ffl 0 =ffl)) steps, where ae is the trace norm of a solution and ffl 0 is the residual error at the (normalized) starting point. A simple way of monitoring possible infeasibility of the original SDP problem is provided such that in at most O( p n ln(aeffl 0 =ffl)) steps either an ffl-approximate solution is obtained, or it is determined that there is no solution with trace norm less than or equal to a given number ae ? 0. Key Words: semidefinite programming, homogeneous interior-point algorithm, polynomial complexity. Abbreviated Title: Homogeneous al...

### Citations

181 | On adaptive-step primal-dual interior-point algorithms for linear programming
- Mizuno, Todd, et al.
- 1993
(Show Context)
Citation Context ...k =�� k ) which does not depend on the magnitude of the starting point. As a final comment we note that we can also extend the long--step algorithm for linear programming by Kojima Mizuno and Yosh=-=ise [2]-=- to the homogeneous system (2.1). By using the proof techniques and the results of Monteiro [5] we can prove that the iteration complexity of this extended algorithm is O(n 1:5 ln(aeffl 0 =ffl)). Refe... |

177 | Primal-dual interior-point methods for self-scaled cones
- Nesterov, Todd
- 1998
(Show Context)
Citation Context ...nsider the primal--dual SDP problem: A i ffl X = b i ; i = 1; : : : ; m; (1.3a) m X i=1 y i A i + S = C; (1.3b) X ffl S = 0; Xs0; Ss0: (1.3c) Recently, Kojima, Shindoh and Hara [4], Nesterov and Todd =-=[8]-=-, and Monteiro [5] extended some interior--point methods from LP to SDP. In the latter paper Monteiro developed a new formulation of the primal--dual search direction originally introduced in [4]. All... |

135 | Primal–dual path-following algorithms for semidefinite programming
- Monteiro
- 1997
(Show Context)
Citation Context ...-dual SDP problem: A i ffl X = b i ; i = 1; : : : ; m; (1.3a) m X i=1 y i A i + S = C; (1.3b) X ffl S = 0; Xs0; Ss0: (1.3c) Recently, Kojima, Shindoh and Hara [4], Nesterov and Todd [8], and Monteiro =-=[5]-=- extended some interior--point methods from LP to SDP. In the latter paper Monteiro developed a new formulation of the primal--dual search direction originally introduced in [4]. All above mentioned m... |

85 |
An O(nL)-iteration homogenous and self-dual linear programming algorithm
- Ye, Todd, et al.
- 1994
(Show Context)
Citation Context ...problem (1.3). This formulation is an extension of the homogeneous and self--dual feasibility model for linear programming developed by Goldman and Tucker [1, 10]. We mention that Ye, Todd and Mizuno =-=[12]-=- proposed a homogeneous algorithm for linear programming, and their results were simplified by Xu, Hung and Ye [11] who developed an efficient code by using Goldman and Tucker's homogeneous formulatio... |

61 |
Interior-point methods for the monotone linear complementarity problem in symmetric matrices
- KOJIMA, SHINDOH, et al.
- 1997
(Show Context)
Citation Context ...arly independent. We consider the primal--dual SDP problem: A i ffl X = b i ; i = 1; : : : ; m; (1.3a) m X i=1 y i A i + S = C; (1.3b) X ffl S = 0; Xs0; Ss0: (1.3c) Recently, Kojima, Shindoh and Hara =-=[4]-=-, Nesterov and Todd [8], and Monteiro [5] extended some interior--point methods from LP to SDP. In the latter paper Monteiro developed a new formulation of the primal--dual search direction originally... |

56 | A simplified homogeneous and self-dual linear programming algorithm and its
- Xu, Hung, et al.
- 1996
(Show Context)
Citation Context ...ramming developed by Goldman and Tucker [1, 10]. We mention that Ye, Todd and Mizuno [12] proposed a homogeneous algorithm for linear programming, and their results were simplified by Xu, Hung and Ye =-=[11]-=- who developed an efficient code by using Goldman and Tucker's homogeneous formulation for linear programming and an infeasible--interior--point method to solve it. Using a similar idea, we apply two ... |

48 | A superlinearly convergent primal-dual infeasible-interiorpoint algorithm for semidefinite programming
- Potra, Sheng
(Show Context)
Citation Context ... conic problems, which include SDP, by trying to find a recession direction of a shifted homogeneous primal--dual problem. More recently, Zhang [13], Kojima, Shida and Shindoh [3] and Potra and Sheng =-=[9]-=- independently proposed new infeasible-interior-point path--following algorithms for SDP. In the present paper, we propose a simple homogeneous primal-dual feasibility model for the problem (1.3). Thi... |

47 | On extending primal-dual interior-point algorithms from linear programming to semidefinite programming
- Zhang
- 1998
(Show Context)
Citation Context ...al infeasible-- interior--point algorithms for nonlinear conic problems, which include SDP, by trying to find a recession direction of a shifted homogeneous primal--dual problem. More recently, Zhang =-=[13]-=-, Kojima, Shida and Shindoh [3] and Potra and Sheng [9] independently proposed new infeasible-interior-point path--following algorithms for SDP. In the present paper, we propose a simple homogeneous p... |

46 | S.: Local convergence of predictor-corrector infeasible-interior-point algorithms for sdps and sdlcps
- Kojima, Shida, et al.
(Show Context)
Citation Context ...algorithms for nonlinear conic problems, which include SDP, by trying to find a recession direction of a shifted homogeneous primal--dual problem. More recently, Zhang [13], Kojima, Shida and Shindoh =-=[3]-=- and Potra and Sheng [9] independently proposed new infeasible-interior-point path--following algorithms for SDP. In the present paper, we propose a simple homogeneous primal-dual feasibility model fo... |

28 |
Polyhedral convex cones
- Goldman, Tucker
(Show Context)
Citation Context ...eneous primal-dual feasibility model for the problem (1.3). This formulation is an extension of the homogeneous and self--dual feasibility model for linear programming developed by Goldman and Tucker =-=[1, 10]-=-. We mention that Ye, Todd and Mizuno [12] proposed a homogeneous algorithm for linear programming, and their results were simplified by Xu, Hung and Ye [11] who developed an efficient code by using G... |

23 | Long-step strategies in interior-point primal-dual methods - Nesterov - 1997 |

3 |
Infeasible start interior point primal-dual methods in nonlinear programming
- NESTEROV
- 1994
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Citation Context ...of the infeasible--interior--point potential-- reduction method of Kojima, Shindoh and Hara [4], require a strictly feasible starting point and therefore are feasible-interior-point methods. Nesterov =-=[7]-=- proposed several infeasible-- interior--point algorithms for nonlinear conic problems, which include SDP, by trying to find a recession direction of a shifted homogeneous primal--dual problem. More r... |

2 |
Dual systems of homogeneous relations
- Tucker
- 1956
(Show Context)
Citation Context ...eneous primal-dual feasibility model for the problem (1.3). This formulation is an extension of the homogeneous and self--dual feasibility model for linear programming developed by Goldman and Tucker =-=[1, 10]-=-. We mention that Ye, Todd and Mizuno [12] proposed a homogeneous algorithm for linear programming, and their results were simplified by Xu, Hung and Ye [11] who developed an efficient code by using G... |