## A simplified homogeneous and self-dual linear programming algorithm and its implementation (1996)

Venue: | Annals of Operations Research |

Citations: | 58 - 5 self |

### BibTeX

@ARTICLE{Xu96asimplified,

author = {Xiaojie Xu and Pi-fang Hung and Yinyu Ye},

title = {A simplified homogeneous and self-dual linear programming algorithm and its implementation},

journal = {Annals of Operations Research},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x

### Citations

297 |
On the implementation of a primal-dual interior point method
- Mehrotra
- 1992
(Show Context)
Citation Context ... problem is infeasible or unbounded, the algorithm correctly detects infeasibility for at least one of the primal and dual problems. Using Mehrotra's simple predictor and corrector strategy (Mehrotra =-=[8]-=-), i.e, taking both predictor and corrector steps with the same matrix factors in one iteration, we develop an efficient implementation code in 1spractice. We propose a technique allowing different st... |

196 | On adaptive primal-dual interior-point algorithms for linear programming
- Mizuno, Todd, et al.
- 1993
(Show Context)
Citation Context ...CA ; \Omega (1) ^ _ ` ^ O(1) 9???= ???; for some fi 2 (0; 1). The boundary property of this type of surface is discussed in Mizuno et al. [11]. Applying Mizuno-Todd-Ye's predictor-corrector technique =-=[10]-=- for (HLF), with fl = 0; and j = 1 (or 0 ! j ^ 1 + O(1=(n + 1)) finite times) (13) in the predictor step, and fl = 1; and j = 0 (or 0 ! j ^ O(1=pn + 1) finite times) (14) in the corrector step, we can... |

95 |
On implementing Mehrotra’s predictor–corrector interior-point method for linear programming
- Lustig, Marsten, et al.
- 1992
(Show Context)
Citation Context ...t in performance for our implementation. Primal and dual step sizes Numerical experience shows that properly taking different step sizes in primal and dual updates may enhance algorithm's performance =-=[5]-=-. We adapt this strategy in our implementation. Let ffx = min f \Gamma xi=(dx)i; if (dx)i ! 0 g; ffz = min f \Gamma zi=(dz)i; if (dz)i ! 0 g; ffo/ = \Gamma o/ =do/ ; if do/ ! 0; ff^ = \Gamma ^=d^; if ... |

66 |
A primal–dual infeasible-interior-point algorithm for linear programming
- Kojima, Megiddo, et al.
- 1993
(Show Context)
Citation Context ...thod directly to solving (HLF). In each iteration, the method basically solves the following system of linear equations for direction (dy; dx; do/ ; dz; d^), as proposed by Kojima, Megiddo and Mizuno =-=[3]-=-, A dx \Gamma b do/ = j rkP ; \Gamma AT dy +c do/ \Gamma dz = \Gamma j rkD; bT dy \Gamma cT dx \Gamma d^ = j rkG; (3) and Xkdz + Zkdx = fl_ke \GammasXkzk; o/ kd^ + ^kdo/ = fl_k \Gammaso/ k^k; (4) wher... |

30 |
Polyhedral convex cones
- Goldman, Tucker
(Show Context)
Citation Context ...linear feasibility model has the form (HLF ) Ax \Gamma bo/ = 0; \Gamma AT y +co/ * 0; bT y \Gamma cT x * 0; y free; x * 0; o/ * 0: (1) This system was first proposed and studied by Goldman and Tucker =-=[2]-=-[12]. Denote by z the slack vector for the second (inequality) constraint and by ^ the slack scalar for the third (inequality) constraint. Then, we are interested in finding a strictly complementary p... |

22 | Finding an Interior Point in the Optimal Face of Linear Programs - Mehrotra, Ye - 1993 |

20 |
Computational experience with a globally convergent primal–dual predictor–corrector algorithm for linear programming
- Lustig, Marsten, et al.
- 1994
(Show Context)
Citation Context ...ty for practical interior-point methods. Some theoretical results hold only for feasible cases. Other approaches to detect infeasibility are somewhat difficult to implement in practice (Lustig et al. =-=[6]-=-). Thus, our method may be a reliable alternative to resolve this issue. 2 A homogeneous and self-dual linear feasibility model The homogeneous and self-dual linear feasibility model has the form (HLF... |

20 |
Dual systems of homogeneous linear relations
- Tucker
- 1956
(Show Context)
Citation Context ...ear feasibility model has the form (HLF ) Ax \Gamma bo/ = 0; \Gamma AT y +co/ * 0; bT y \Gamma cT x * 0; y free; x * 0; o/ * 0: (1) This system was first proposed and studied by Goldman and Tucker [2]=-=[12]-=-. Denote by z the slack vector for the second (inequality) constraint and by ^ the slack scalar for the third (inequality) constraint. Then, we are interested in finding a strictly complementary point... |

14 |
An O(pnL)-iteration homogeneous and self-dual linear programming algorithm
- Ye, Todd, et al.
- 1994
(Show Context)
Citation Context ...nly if it is feasible and bounded. The dual problem of (LP) can be written as (LD) maximize bT y subject to AT y ^ c; where y 2 Rm. We call z = c \GammasAT y 2 Rn dual slacks. Recently Ye-Todd-Mizuno =-=[15]-=- developed a homogeneous and self-dual (HLP) linear programming algorithm based on the construction of a homogeneous and self-dual (artificial) LP model, in which the dimension of the problem is incre... |

11 |
order methods and their performance
- Mehrotra, High
- 1990
(Show Context)
Citation Context ...az + fl_ke; o/ kd^ + ^kdo/ = \Gamma o/ k^k \Gammasdao/ da^ + fl_k: 8s(Note that the system is not exactly the centering step since it contains Daxdaz and dao/ da^ on the right-hand side, see Mehrotra =-=[7]-=-[8].) For system (3), we choose j = 1 \Gammasfl in the corrector step and j = 1 in the predictor step. Solving normal equation system The major part of computational work in each iteration is concerne... |

8 |
An efficient implementation of a higher order primal dual interior point method for large scale linear programming
- Altman, Gondzio
- 1992
(Show Context)
Citation Context ...rom a simple initial point such as x = e, y = 0, z = e. Several papers considered various initial point selections, see Lustig et al. [4], Lustig et al. [5], Mehrotra [7], Mehrotra [8], Altman et al. =-=[1]-=-, and etc. According to their descriptions, they generally need to solve a least-squares problem at the beginning to construct the initial point, where the amount of work is like one iteration. Howeve... |

5 |
A Surface of Analytic Centers and Infeasible-Interior-Point Algorithms for Linear Programming
- MIZUNO, TODD, et al.
- 1995
(Show Context)
Citation Context ...rates converge to the all-zero solution which is the origin of the cone (HLF). Furthermore, if _k=`k ! 1, then we have diverging iterates. This behavior has been rigorously discussed by Mizuno et al. =-=[11]-=-. Nevertheless, while maintaining relations (9) and (12), we have some freedom in choosing j and fl in each iteration. In particular, with j = 1 \Gammasfl in all iterations, the generic algorithm beco... |

3 |
Starting and restarting the primal–dual interior point method
- Lustig, Marsten, et al.
- 1990
(Show Context)
Citation Context ...r them to solve some of the NETLIB feasible problems if starting from a simple initial point such as x = e, y = 0, z = e. Several papers considered various initial point selections, see Lustig et al. =-=[4]-=-, Lustig et al. [5], Mehrotra [7], Mehrotra [8], Altman et al. [1], and etc. According to their descriptions, they generally need to solve a least-squares problem at the beginning to construct the ini... |

2 |
On quadratic convergence of the homogeneous and self-dual linear programming algorithm. Working paper
- Wu, Wu, et al.
- 1993
(Show Context)
Citation Context ...Lustig et al. [5] and Mehrotra [8]. For solving infeasible LP problems, we notice that o/ k=^k does decrease quadratically in our experiment, which correctly proves infeasibility in theory (Wu et al. =-=[13]-=-). We find that the use of criterion o/ ^ = o/ 0 ^0 ! 10 \Gamma 8 or o/ ^ = o/ 0 ^0 ! 10 \Gamma 10 ususally does not make a difference in running time. Initial point We choose the starting point simpl... |

1 |
Interior Point Method for Linear Programming : Theory
- Xu
- 1991
(Show Context)
Citation Context ...h attempts to minimize the number of nonzeros in L. The factorization procedure is written in FORTRAN 77, based on the framework of IPMOS, an interior-point method optimization system developed by Xu =-=[14]-=-. Memory allocation is carefully managed for exploiting cache memory. During the factorization, if a pivot value is less than a machine related tolerance, then the current row i can be regarded as an ... |