## Infeasible-Start Primal-Dual Methods And Infeasibility Detectors For Nonlinear Programming Problems (1996)

Venue: | Mathematical Programming |

Citations: | 34 - 5 self |

### BibTeX

@TECHREPORT{Nesterov96infeasible-startprimal-dual,

author = {Yu Nesterov and M. J. Todd and Y. Ye},

title = {Infeasible-Start Primal-Dual Methods And Infeasibility Detectors For Nonlinear Programming Problems},

institution = {Mathematical Programming},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we present several "infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a self-dual homogeneous primal-dual problem. The methods under consideration generate an ffl-solution for an ffl-perturbation of an initial strictly (primal and dual) feasible problem in O( p ln fflae f ) iterations, where is the parameter of a self-concordant barrier for the cone, ffl is a relative accuracy and ae f is a feasibility measure. We also discuss the behavior of path-following methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O( p ln ae \Delta ) iterations, where ae \Delta is a primal or dual infeasibility measure. 1 Introduction Nesterov and Nemirovskii [9] first developed and investigated extensions of several classes of interior-point algorithms for linear programming t...

### Citations

192 | Primal-dual interior-point methods for self-scaled cones
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Citation Context ...ain, with a suitable starting point, the estimate ��(��x k ; y k ; �� s k )sexp ae ` ps+ 1 k oe (5.10) for all ks0. 5.4 Short-step path-following methods Here we show that the methods of S=-=ection 6 of [11] can be used to increase ��(��x;-=- y; �� s) at each iteration. We suppose that our current iterate (��x k ; y k ; �� s k ) lies in the narrow neighborhood N (fi) := f(��x; y; �� s) 2 Q : k��s=��(��x; y;... |

183 |
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Citation Context ...gnores the primal-dual symmetry in the vector z and the barrier \Phi, while the second exploits these features and the skewsymmetry of G, but requires that the barrier F and the cone K be self-scaled =-=[10, 11], so tha-=-t �� F and �� K are also. We will give the definition of this concept later, when we introduce these structured methods. For now, we consider only the general methods. As in [7], [8] the effic... |

93 | Linear Programming, Complexity Theory and Elementary Functional Analysis
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Citation Context ... to solving "self-scaled" convex problems. Most of these results assume that the initial problem and its dual have strictly interior feasible points. One paper that relaxes this assumption i=-=s Renegar [15]-=-, which studies the complexity of using primal barrier methods and a two-phase approach to first determine feasibility and then, if feasible, obtain approximate optimality. The aim of this paper is tw... |

93 |
An O( √ nL)–iteration homogeneous and self–dual linear programming algorithm
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- 1994
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Citation Context ...approach to first determine feasibility and then, if feasible, obtain approximate optimality. The aim of this paper is twofold. First, we extend the homogeneous and self-dual linear programming model =-=[18]-=- to convex programming problems in conic form. In this way, we are able to start an interior-point algorithm without any prior knowledge of the feasibility (primal or dual) of the initial problem. (Ea... |

76 | Some perturbation theory for linear programming
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Citation Context ...r, under suitable conditions we obtain these indications in O( psln ffl ) iterations of certain algorithms. Thus, our complexity bounds hold even for some "ill-posed" problems, a term used i=-=n Renegar [13, 14]. Furtherm-=-ore, if the primaldual pair is strictly primal or dual infeasible, we can obtain an "exact" certificate of strict primal or dual infeasibility in O( psln ae \Delta ) iterations, where ae \De... |

52 | Incorporating condition measures into the complexity theory of linear programming
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- 1995
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Citation Context ...r, under suitable conditions we obtain these indications in O( psln ffl ) iterations of certain algorithms. Thus, our complexity bounds hold even for some "ill-posed" problems, a term used i=-=n Renegar [13, 14]. Furtherm-=-ore, if the primaldual pair is strictly primal or dual infeasible, we can obtain an "exact" certificate of strict primal or dual infeasibility in O( psln ae \Delta ) iterations, where ae \De... |

47 |
Centered Newton method for mathematical programming
- Tanabe
- 1988
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Citation Context ...+ 1 \Gamma 1, and then we easily find P fl (u) = ( + 1 \Gamma ps+ 1)(ln ��(u) + 1) + F (x) + Fs(s) \Gamma ln �� \Gamma lns\Gamma 1: This is very similar to the standard primal-dual potential f=-=unction [16, 17], but wi-=-th the expected coefficient ( + 1 + ps+ 1) of ln ��(u) replaced by ( + 1 \Gamma ps+ 1); this change is appropriate because here we seek to increase, rather than decrease, ��(u). Consider a New... |

41 |
A centered projective algorithm for linear programming
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- 1990
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Citation Context ...+ 1 \Gamma 1, and then we easily find P fl (u) = ( + 1 \Gamma ps+ 1)(ln ��(u) + 1) + F (x) + Fs(s) \Gamma ln �� \Gamma lns\Gamma 1: This is very similar to the standard primal-dual potential f=-=unction [16, 17], but wi-=-th the expected coefficient ( + 1 + ps+ 1) of ln ��(u) replaced by ( + 1 \Gamma ps+ 1); this change is appropriate because here we seek to increase, rather than decrease, ��(u). Consider a New... |

37 | Initialization in semidefinite programming via a self-dual skew-symmetric embedding
- Klerk, Roos, et al.
- 1997
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Citation Context ...start an interior-point algorithm without any prior knowledge of the feasibility (primal or dual) of the initial problem. (Earlier work on this topic includes Nesterov [8], de Klerk, Roos and Terlaky =-=[5]-=-, Luo, Sturm and Zhang [6], Potra and Sheng [12], and Andersen and Ye [1].) We present several path-following and potential-reduction primal-dual interior-point methods, which try to find a recession ... |

36 | On homogeneous interior-point algorithms for semidefinite programming - Potra, Sheng - 1998 |

32 | On a homogeneous algorithm for the monotone complementarity problem
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- 1999
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Citation Context ...ibility (primal or dual) of the initial problem. (Earlier work on this topic includes Nesterov [8], de Klerk, Roos and Terlaky [5], Luo, Sturm and Zhang [6], Potra and Sheng [12], and Andersen and Ye =-=[1].) We pres-=-ent several path-following and potential-reduction primal-dual interior-point methods, which try to find a recession direction of the feasible set of a "projective version" of the model. Suc... |

27 |
Long-step strategies in interior-point primal-dual methods
- Nesterov
- 1997
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Citation Context ...t the optimal set of the primal-dual problem min x;y;s hc; xi \Gamma hb; yi (PD) s.t. Ax = b; A T y + s = c; x 2 K; s 2 K ; (2.15) is nonempty and bounded and that its optimal value is zero (see [9], =-=[7]-=-). Whether Assumption 2 holds or not, it is easy to see that if x and (y; s) are feasible in (P) and (D) respectively, then hc; xishb; yi, so if equality holds both are optimal, and the pair is optima... |

25 |
A polynomial Newton method for linear programming
- Ghellinck, Vial
- 1986
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Citation Context ...rivial recession directions exist only for an unbounded set and that is exactly the case when a self-concordant barrier does not achieve its minimum.) This idea was used first by de Ghellink and Vial =-=[2]-=- for solving a system of linear inequalities. It was also used in [7] for constructing adaptive-step path-following methods for NLP and in [8] for developing infeasible-start methods for NLP. However,... |

24 |
Hertog, Interior Point Approach to Linear
- den
- 1994
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Citation Context ...gated extensions of several classes of interior-point algorithms for linear programming to general convex programming. Many others have also studied methods of this kind: see, for example, den Hertog =-=[3] and -=-Jarre [4] and the references therein. More recently, Nesterov and Todd [10, 11] have developed a theoretical foundation for the extension of a popular symmetric primal-dual algorithm to solving "... |

20 | Duality and self-duality for conic convex programming
- Luo, Sturm, et al.
- 1996
(Show Context)
Citation Context ...gorithm without any prior knowledge of the feasibility (primal or dual) of the initial problem. (Earlier work on this topic includes Nesterov [8], de Klerk, Roos and Terlaky [5], Luo, Sturm and Zhang =-=[6]-=-, Potra and Sheng [12], and Andersen and Ye [1].) We present several path-following and potential-reduction primal-dual interior-point methods, which try to find a recession direction of the feasible ... |

16 | Condition number complexity of an elementary algorithm for resolving a conic linear system
- Epelman, Freund
- 1997
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Citation Context ... which studies the complexity of using primal barrier methods and a two-phase approach to first determine feasibility and then, if feasible, obtain approximate optimality. Further, Epelman and Freund =-=[2]-=- investigate the complexity of an elementary algorithm for finding a near-feasible point for a general conic system or proving infeasibility. The aim of this paper is twofold. First, we extend the hom... |

12 |
Interior-point Methods via Self-concordance or Relative Lipschitz Condition
- Jarre
- 1994
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Citation Context ...ns of several classes of interior-point algorithms for linear programming to general convex programming. Many others have also studied methods of this kind: see, for example, den Hertog [3] and Jarre =-=[4] and the r-=-eferences therein. More recently, Nesterov and Todd [10, 11] have developed a theoretical foundation for the extension of a popular symmetric primal-dual algorithm to solving "self-scaled" c... |

3 |
Infeasible start interior point primal-dual methods in nonlinear programming
- Nesterov
- 1994
(Show Context)
Citation Context ... 5 There exists an absolute constant ` ? 0 depending only on \Delta 0 , \Delta 1 and \Delta 2 , such that ��(u k )sexp ae ` ps+ 1 k oe (4.11) for any ks0. The proof is similar to that of Theorem 4=-= in [8]-=-. 5 Primal-dual infeasible-start methods, II In this section we describe methods that use the primal-dual symmetry in \Phi and Q but are based on the ideas of the previous section. In order to exploit... |