## in a wide neighborhood of the central path

### BibTeX

@MISC{Potra_ina,

author = {Florian A. Potra and Xing Liu},

title = {in a wide neighborhood of the central path},

year = {}

}

### OpenURL

### Abstract

Copyright information to be inserted by the Publishers Predictor-corrector methods for sufficient linear complementarity problems

### Citations

507 | Primal-Dual Interior-Point Methods
- Wright
- 1997
(Show Context)
Citation Context ...g (QP), semidefinite programming (SDP), and many other problems. For an excellent analysis of primal-dual interior-point methods and their implementation see the excellent monograph of Stephen Wright =-=[24]-=-. The MTY predictor-corrector algorithm proposed by Mizuno, Todd and Ye [12] is a typical representative of a primal-dual interior-point method for LP. It has O( √ nL) iteration complexity, which is t... |

202 | On adaptive-step primal-dual interior-point algorithms for linear programming
- Mizuno, Todd, et al.
- 1993
(Show Context)
Citation Context ...llent analysis of primal-dual interior-point methods and their implementation see the excellent monograph of Stephen Wright [24]. The MTY predictor-corrector algorithm proposed by Mizuno, Todd and Ye =-=[12]-=- is a typical representative of a primal-dual interior-point method for LP. It has O( √ nL) iteration complexity, which is the best iteration complexity obtained so far for any interior-point method. ... |

159 | A unified approach to interior point algorithms for linear complementarity problems
- Kojima, Megiddo, et al.
- 1991
(Show Context)
Citation Context ...exity and superlinear convergence has recently been proposed in [15]. The existence of a central path is crucial for interior-point methods. An important result of the 1991 monograph of Kojima et al. =-=[9]-=- shows that the central path exists for any P∗ linear complementarity problem, provided that the relative interior of its feasible set is nonempty. We recall that every P∗ linear complementarity probl... |

54 |
On quadratic and O( √ nL) convergence of a predictor-corrector algorithm for LCP
- Anstreicher, Ye
- 1993
(Show Context)
Citation Context ...onvergence with factor at most .25. A direct proof of the superlinear convergence of the MTY algorithm for nondegenerate LCPs, without using the convergence of the iteration sequence, is contained in =-=[26]-=-. The MTY algorithm operates in a l2 neighborhood of the central path. It is well known however that primal-dual interior-point methods have a better practical performance in a wider neighborhood of t... |

44 | On the local convergence of a predictor-corrector method for semidefinite programming
- Ji, Potra, et al.
- 1999
(Show Context)
Citation Context ...t appears that they are not superlinearly convergent. A higher order algorithm of MTY type in the N − ∞ neighborhood with O( √ nL) complexity and superlinear convergence has recently been proposed in =-=[15]-=-. The existence of a central path is crucial for interior-point methods. An important result of the 1991 monograph of Kojima et al. [9] shows that the central path exists for any P∗ linear complementa... |

37 |
A quadratically convergent O( p nL)- iteration algorithm for linear programming
- Ye, Guler, et al.
- 1993
(Show Context)
Citation Context ...plexity, which is the best iteration complexity obtained so far for any interior-point method. Moreover, the duality gap of the sequence generated by the MTY algorithm converges to zero quadratically =-=[27]-=-. The MTY algorithm was the first algorithm for LP having both polynomial complexity and ∗ Work supported in part by the National Science Foundation, Grant No. 0139701. 12 F. A. Potra AND X. Liu supe... |

36 | Local convergence of interior-point algorithms for degenerate monotone LCPs
- Monteiro, Wright
- 1993
(Show Context)
Citation Context ...te and the iteration sequence converges. It turns out that these assumptions are not restrictive. Thus the convergence of the iteration sequence follows from a general result of [4] and, according to =-=[13]-=-, the nondegeneracy (i.e. the existence of a strict complementarity solution) is a necessary condition for superlinear convergence. We note that in [13] it is shown that in the degenerate case a large... |

33 |
A predictor-corrector method for linear complementarity problems with polynomial complexity and superlinear convergence
- Ji, Potra, et al.
- 1991
(Show Context)
Citation Context ...hm for LP having both polynomial complexity and ∗ Work supported in part by the National Science Foundation, Grant No. 0139701. 12 F. A. Potra AND X. Liu superlinear convergence. Ji, Potra and Huang =-=[8]-=- generalized the MTY algorithm to monotone linear complementarity problems (LCP). The method has O( √ nL) iteration complexity and superlinear convergence, under the assumption that the LCP is non-deg... |

27 |
Self Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms
- Peng, Roos, et al.
- 2002
(Show Context)
Citation Context ...lts4 F. A. Potra AND X. Liu are given. All the above mentioned interior-point methods for sufficient linear complementarity problem use small neighborhoods of the central path. In a recent monograph =-=[7]-=-, Peng, Roos and Terlaky propose the use of larger neighborhoods of the central path defined by means of self-regular functions. They propose an interior-point algorithm for solving a class of P∗(κ) n... |

24 | Convergence of interior point algorithms for the monotone linear complementarity problem
- Bonnans, Gonzaga
- 1996
(Show Context)
Citation Context ...he LCP is non-degenerate and the iteration sequence converges. It turns out that these assumptions are not restrictive. Thus the convergence of the iteration sequence follows from a general result of =-=[4]-=- and, according to [13], the nondegeneracy (i.e. the existence of a strict complementarity solution) is a necessary condition for superlinear convergence. We note that in [13] it is shown that in the ... |

24 |
An asymptotical O( √ nL)-iteration path-following linear programming algorihtm that used wide neighborhoods
- Huang, Ye
- 1996
(Show Context)
Citation Context ...ictor-corrector methods in large neighborhoods. The best iteration complexity achieved by any known interior-point method in the large neighborhood using first order information is O(nL). As shown in =-=[6, 28]-=-, the iteration complexity can be reduced to O( √ nL) by using higher order information. However the algorithms presented in those papers are not of MTY type and it appears that they are not superline... |

22 | A large-step infeasible-interior-point method for the P∗matrix LCP - Potra, Sheng - 1997 |

17 |
High order infeasible-interior-point methods for solving sufficient linear complementarity problems
- Stoer, Wechs, et al.
- 1998
(Show Context)
Citation Context ...erate the convergence of the corresponding variables. Predictor-corrector algorithms with arbitrarily high order of convergence for degenerate sufficient linear complementarity problems were given in =-=[20]-=-. The algorithms depend on the constant κ, use a l2 neighborhood of the central path and, as shown in [19], have O((1 + κ) √ nL) iteration complexity for P∗(κ) linear complementarity problems. A gener... |

16 | Interior-point algorithms for linear complementarity problems based on large neighborhoods of the central path
- Zhao
- 1998
(Show Context)
Citation Context ...ictor-corrector methods in large neighborhoods. The best iteration complexity achieved by any known interior-point method in the large neighborhood using first order information is O(nL). As shown in =-=[6, 28]-=-, the iteration complexity can be reduced to O( √ nL) by using higher order information. However the algorithms presented in those papers are not of MTY type and it appears that they are not superline... |

15 |
A superlinearly convergent infeasible-interior-point algorithm for geometrical LCPs without a strictly complementary condition
- Mizuno
- 1994
(Show Context)
Citation Context ... that they are superlinearly convergent for degenerate problems does not contradict the result of that paper. In the degenerate case superlinear convergence is achieved by employing an idea of Mizuno =-=[11]-=- which consists in identifying indices for which strict complementarity does not hold, which is possible when the complementarity gap is small enough, and by using an extra backsolve in order to accel... |

15 | Superlinearly convergent infeasible–interior–point algorithm for degenerate LCP
- Potra, Sheng
- 1998
(Show Context)
Citation Context ... κ, their computational complexity does: if the problem is a P∗(κ) linear complementarity problem they terminate in at most O((1 + κ) √ nL) iterations. The predictor-corrector algorithms presented in =-=[17, 18]-=- are superlinearly convergent even for degenerate problems. More precisely the Q-order of convergence of the complementarity gap is 2 for nondegenerate problems and 1.25 for degenerate problems. The a... |

15 |
Infeasible-interior-point paths for sufficient linear complementarity problems and their analyticity
- Stoer, Wechs
- 1998
(Show Context)
Citation Context ... order of convergence for degenerate sufficient linear complementarity problems were given in [20]. The algorithms depend on the constant κ, use a l2 neighborhood of the central path and, as shown in =-=[19]-=-, have O((1 + κ) √ nL) iteration complexity for P∗(κ) linear complementarity problems. A general local analysis of higher order predictor-corrector methods in a l2 neighborhood for degenerate sufficie... |

15 |
P∗ matrices are just sufficient
- Väliaho
- 1996
(Show Context)
Citation Context ...oincides with the class of monotone linear complementarity prob-A wide neighborhood predictor-corrector method for LCP 3 lems, and 0 ≤ κ1 ≤ κ2 implies P∗(κ1) ⊂ P∗(κ2). A surprising result of Väliaho =-=[22]-=- from 1996 showed that the class of P∗ matrices coincides with the class of sufficient matrices. Therefore, the interior-point methods of [9] can solve any sufficient linear complementarity problem. O... |

15 | A superquadratic infeasible-interior-point method for linear complementarity problems
- Wright, Zhang
- 1994
(Show Context)
Citation Context ...ing a higher order predictor, the algorithms of [16] may attain arbitrarily high orders of convergence on nondegenerate problems. Finally, by using the ”fastsafe-improve” strategy of Wright and Zhang =-=[25]-=-, the algorithms of [16] require asymptotically only one matrix factorization per iteration, while Miao’s algorithm, as well as the original MTY algorithm, require two matrix factorizations at every i... |

13 |
Complexity of predictor-corrector algorithms for LCP based on a large neighborhood of the central path
- Gonzaga
(Show Context)
Citation Context ...r and Bosch [3] it follows that the iteration complexity of a straightforward implementation of a predictor-corrector method in the large neighborhood of the central path would be O(n 3/2 L). Gonzaga =-=[5]-=- proposed a predictor-corrector method using the N∞ neighborhood of the central path that has O(nL) iteration complexity. In contrast with the MTY algorithm that uses a predictor step followed by a co... |

12 | An infeasible-interior-point predictor-corrector algorithm for the P∗-geometric
- Anitescu, Lesaja, et al.
- 1997
(Show Context)
Citation Context ...ence, including the MTY predictor-corrector method [4, 13]. The following result was proved for standard monotone LCP in [26] and for HLCP in [4]. Its extension for P∗(κ) HLCP is straightforward (see =-=[2]-=-). LEMMA 3.5. If F # ̸= ∅ then the solution w = ⌈ u, v ⌋ of ( 3.10) satisfies where µ = µ(z) is given by (1.2). | uivi | = O(µ 2 ), ∀i ∈ {1, 2, . . . , n} ,16 F. A. Potra AND X. Liu With the help of ... |

11 | F.A.: Equivalence between different formulations of the linear complementarity problems
- Anitescu, Lesaja, et al.
- 1997
(Show Context)
Citation Context ...CP) in the N − ∞ neighborhood of the central path that extends the algorithm of [15]. HLCP is a slight generalization of the standard linear complementarity problem (LCP). As shown by Anitescu et al. =-=[1]-=-, different variants of the P∗(κ) linear complementarity problem, including LCP, HLCP, mixed LCP and geometric LCP are equivalent in the sense that any complexity or superlinear convergence result pro... |

11 |
A new infinity-norm path following algorithm for linear programming
- Anstreicher, Bosch
- 1995
(Show Context)
Citation Context ...exity of the predictor-corrector methods that use wide neighborhoods are worse than the complexity of the corresponding methods for small neighborhoods. By using the analysis of Anstreicher and Bosch =-=[3]-=- it follows that the iteration complexity of a straightforward implementation of a predictor-corrector method in the large neighborhood of the central path would be O(n 3/2 L). Gonzaga [5] proposed a ... |

10 |
An O(nL) infeasible interior–point algorithm for LCP with quadratic convergence
- Potra
- 1996
(Show Context)
Citation Context ...we obtain Du 1 + D −1 v 1 = −(1 + ɛ)(xs) 1/2 j=1 Du 2 + D −1 v 2 = ɛ(xs) 1/2 − (xs) −1/2 u 1 v 1 Du i + D −1 v i = −(xs) −1/2 i−1 Du j D −1 v i−j , i = 3, . . . , m . Using Lemma3.2, Corollary 2.3 of =-=[14]-=-, and the fact z ∈ D(β), we deduce that η1 = (1 + ɛ) √ nµ , ∑ j=1 η 2 2 ≤ ɛ 2 nµ − 2ɛ(u 1 ) T (v 1 ) + ‖u1 v 1 ‖ 2 2 βµ ) i ,20 F. A. Potra AND X. Liu ≤ ɛ 2 nµ + 2ɛκη 2 1 + 1 8βµ (1 + 4κ + 8κ2 )η 4 1... |

10 | A path following method for LCP with superlinearly convergent iteration sequence
- Potra, Sheng
- 1998
(Show Context)
Citation Context ... κ, their computational complexity does: if the problem is a P∗(κ) linear complementarity problem they terminate in at most O((1 + κ) √ nL) iterations. The predictor-corrector algorithms presented in =-=[17, 18]-=- are superlinearly convergent even for degenerate problems. More precisely the Q-order of convergence of the complementarity gap is 2 for nondegenerate problems and 1.25 for degenerate problems. The a... |

8 |
A quadratically convergent O((1 + ) p nL)-iteration algorithm for the P ()- matrix linear complementarity problem
- Miao
- 1993
(Show Context)
Citation Context ...he best known iteration complexity of an interior-point method for a P∗(κ) problem is O((1 + κ) √ nL). No superlinear complexity results were given for the interior-point methods of [9]. In 1995 Miao =-=[10]-=- extended the MTY predictor-corrector method for P∗(κ) linear complementarity problems. His algorithm uses the l2 neighborhood of the central path, has O((1 + κ) √ nL) iteration complexity, and is qua... |

6 |
High order long-step methods for solving linear complementarity problems
- Stoer
- 2001
(Show Context)
Citation Context ...given constant, their algorithm is only linearly convergent. A superlinear interior-point algorithm for sufficient linear complementarity problems in the N − ∞ neighborhood has been proposed by Stoer =-=[21]-=-. This algorithm is an adaptation of the second algorithm of [29] for the large neighborhood. No complexity results are proved. In the present paper we will propose several predictor-corrector methods... |

4 | On the rate of local convergence of High-Order-InfeasiblePath-Following Algorithms for P∗-Linear Complementarity Problems
- Zhao, Sun
(Show Context)
Citation Context ... complementarity problems. A general local analysis of higher order predictor-corrector methods in a l2 neighborhood for degenerate sufficient linear complementarity problems is given by Zhao and Sun =-=[29]-=-, who also propose a new algorithm that does not need a corrector step. The latter algorithm does not follow the traditional central path. Instead a new analytic path is used at each iteration. No com... |

2 |
Determining the handicap of a sufficient matrix
- Väliaho
- 1997
(Show Context)
Citation Context ...f a sufficient pair (Q, R) is defined as χ(Q, R) := min{κ : κ ≥ 0, (Q, R) ∈ P∗(κ)} . (2.2) A general expression for the handicap of a sufficient matrix and a method for determining it is described in =-=[23]-=-. We denote the set of all feasible points of HLCP by and its solution set by F = {z = ⌈ x, s ⌋ ∈ IR 2n + : Qx + Rs = b}, F ∗ = {z ∗ = ⌈ x ∗ , s ∗ ⌋ ∈ F : x ∗ s ∗ = 0}.A wide neighborhood predictor-c... |