## Primal-dual interior-point methods (1997)

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@INPROCEEDINGS{Potra97primal-dualinterior-point,

author = {Florian A. Potra and Stephen J. Wright},

title = {Primal-dual interior-point methods},

booktitle = {},

year = {1997},

publisher = {SIAM}

}

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### Abstract

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, suchasconvex quadratic programming, semide nite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semide nite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions. 1

### Citations

934 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
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Citation Context ... optimization problems. In what follows, we will describe the technique of Goemans and Williamson, which yields an approximate solution whose value is within 13% of optimality for the MAX CUT problem =-=[7]-=-. In MAX CUT, we are presented with an undirected graph with N whose edges wij have nonnegative weights. The problem is choose a subsetSf1; 2;:::;Ng so that the sum of weights of the edges that cross ... |

651 | A New Polynomial-Time Algorithm for Linear Programming
- Karmarkar
- 1984
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Citation Context ...ethods had been considered one way or another from the 1950's, and investigated quite extensively during the 1960s (Fiacco and McCormick [5]), it was the publication of the seminal paper of Karmarkar =-=[11]-=- that placed interior-point methods at the top of the agenda for many researchers. On the theoretical side, subsequent research led to improved computational complexity bounds for linear programming (... |

314 |
Nonlinear Programming: Sequential Unconstrained Minimization Techniques
- Fiacco, McCormick
- 1968
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Citation Context ...matical programming in 1990s. Although various interior-point methods had been considered one way or another from the 1950's, and investigated quite extensively during the 1960s (Fiacco and McCormick =-=[5]-=-), it was the publication of the seminal paper of Karmarkar [11] that placed interior-point methods at the top of the agenda for many researchers. On the theoretical side, subsequent research led to i... |

269 |
On the implementation of a primal–dual interior point method
- Mehrotra
- 1992
(Show Context)
Citation Context ...s to convex programming and linear complementarity. In 1989, Mehrotra described a practical algorithm for linear programming that remains the basis of most current software; his work appeared in 1992 =-=[14]-=-. Meanwhile, Nesterov and Nemirovskii [16] were developing the theory of self-concordant functions, which allowed algorithms based on the primal log-barrier function for linear programming to be exten... |

219 |
Interior Point Algorithms: Theory and Analysis
- Ye
- 1997
(Show Context)
Citation Context ... the bibliography of the technical report by Wright [28]. A great deal of literature is available to the reader interested in learning more about interior-point methods. Anumber of recent books [27], =-=[29]-=-, [23] give overviews of the area, from rst principles to new results and practical considerations. Theoretical background on self-concordant functionals and related 2sdevelopments is described in [16... |

175 | Primal–dual interior–point methods for self–scaled cones
- Nesterov, Todd
- 1998
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Citation Context ...primal log-barrier function for linear programming to be extended to wider classes of convex problems, particularly semide nite programming and second-order cone programming (SOCP). Nesterov and Todd =-=[17, 18]-=- extended the primal-dual approach along similar lines to a more restricted class of convex problems that still included SDP and SOCP. Other work on interior-point algorithms for SDPs, which have a wi... |

162 | Self-scaled barriers and interior-point methods in convex programming - Nesterov, Todd - 1997 |

146 |
Pathways to the optimal set in linear programming
- Megiddo
- 1989
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Citation Context ...bsets of the feasible set, each containing the solution, and used Newton's method to follow the analytic centers of these subsets to the primal optimum. A new era was inaugurated with Megiddo's paper =-=[13]-=-, originally presented in 1987, which described a framework for primal-dual framework algorithms. The primal-dual viewpoint proved to be extremely productive. It yielded new algorithms with interestin... |

134 | A Mathematical View of Interior-Point Methods in Convex Optimization
- RENEGAR
- 1999
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Citation Context ... give overviews of the area, from rst principles to new results and practical considerations. Theoretical background on self-concordant functionals and related 2sdevelopments is described in [16] and =-=[22]-=-. Technical reports from the past ve years can be obtained from the Interior-Point Methods Online Web site at www.mcs.anl.gov/otc/InteriorPoint. 2 Linear Programming We consider rst the linear program... |

121 |
A polynomial-time algorithm, based on Newton’s method, for linear programming
- Renegar
- 1988
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Citation Context ... primal problem, but were more amenable to implementation than the original method or that had better complexity bounds. A particularly notable contribution from this period was Renegar's 1salgorithm =-=[21]-=-, which used upper bounds on the optimal objectivevalue to form successively smaller subsets of the feasible set, each containing the solution, and used Newton's method to follow the analytic centers ... |

88 |
An O( √ nL)-iteration homogeneous and selfdual linear programming algorithm
- Ye, Todd, et al.
- 1994
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Citation Context ...e points exist. They also can be used to detect the infeasibility of certain linear programming problems. A di erent way of dealing with infeasible starting points was proposed by Ye, Todd and Mizuno =-=[31]-=-. Starting with a linear programming problem in standard form and with a possibly infeasible starting point whose x and s components are strictly positive, they construct a homogeneous self-dual linea... |

87 |
Theory and algorithms for linear optimization. an interior point approach
- Vial
- 1997
(Show Context)
Citation Context ...ibliography of the technical report by Wright [28]. A great deal of literature is available to the reader interested in learning more about interior-point methods. Anumber of recent books [27], [29], =-=[23]-=- give overviews of the area, from rst principles to new results and practical considerations. Theoretical background on self-concordant functionals and related 2sdevelopments is described in [16] and ... |

80 | Interior methods for constrained optimization - Wright - 1992 |

51 |
The Simplex Method, A Probabilistic Analysis
- Borgwardt
- 1987
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Citation Context ...omial (that is, a polynomial in the dimension n of the problem only), which is closer to practical experience with this method than the exponential complexity ofthe worst-case analysis (see Borgwardt =-=[3]-=- and the literature cited therein). As mentioned above, the worstcase complexity ofinterior-point methods is weakly polynomial, in the sense that the iteration bounds are polynomials in the dimension ... |

48 | Multiple centrality corrections in a primal–dual method for linear programming
- Gondzio
- 1996
(Show Context)
Citation Context ...binary length of the data. Most interior-point software for linear programming is based on Mehrotra's predictor-corrector algorithm [14], often with the higher-order enhancements described by Gondzio =-=[9]-=-. This approach uses an adaptive choice of k, selected by rst solving for the pure Newton step (that is, setting r = 0 and = 0 in (2.10)). If this step makes good progress in reducing ,wechoose k smal... |

47 | Local Convergence of Predictor-Corrector Infeasible-Interior-Point Method for SDPs and SDLCPs
- Kojima, Shida, et al.
- 1998
(Show Context)
Citation Context ...ear convergence rate indicated by the analysis. 13sThe local convergence analysis for interior-point algorithms for SDP is much more challenging than for linear programming. Kojima, Shida and Shindoh =-=[12]-=- established superlinear convergence of the MizunoTodd-Ye predictor-corrector algorithm based on the KSH/HRVW/M search direction under the following three assumptions: (A) SDP has a strictly complemen... |

39 | A quadratically convergent predictor-corrector method for solving linear programs from infeasible starting points - Potra - 1994 |

36 |
A study of indicators for identifying zero variables in interior-point methods
- El-Bakry, Tapia, et al.
- 1994
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Citation Context ... projection can be used to identify an exact solution. A nite-termination strategy can also be implemented by using the Tapia indicators to decide which components of x and s are zero at the solution =-=[4]-=-. The use of a nite-termination strategy in conjunction with superlinearly convergent algorithms for linear programming is somewhat super uous, since the domain range of k values for which superlinear... |

36 |
Semidefinite programming
- Todd
(Show Context)
Citation Context ... theoretical computational complexityofinterior-point methods for LP was eventually lowered to O( p nL) iterations, requiring a total of O(n3L) bit operations by anumber of authors. Goldfarb and Todd =-=[8]-=- provide a good reference for these complexity results. By using fast matrix multiplication techniques, the complexity estimates can be reduced further. Quite recently, Anstreicher [2] proposed an int... |

36 |
A quadratically convergent O( p nL)-iteration algorithm for linear programming
- Ye, uler, et al.
- 1993
(Show Context)
Citation Context ...or at each step, i.e., k+1 = 1, 0:4 n k; k =0; 1; 2;::: (2.14) the predictor-corrector algorithm, by its adaptive choice of k, allows k to decrease faster, especially close to the solution. Ye et al. =-=[30]-=- proved that the predictor-corrector algorithm is quadratically convergent in the sense that k+1 B 2 k; k=0; 1; 2;::: (2.15) for some constant B independent ofk. This constant may be large, so that (2... |

35 | Semide nite programming - Vandenberghe, Boyd - 1996 |

30 | A study of search directions in primal-dual interior-point methods for semidefinite programming
- Todd
- 1999
(Show Context)
Citation Context ...e Newton's method can be applied the domain and range have to be reconciled. The various primal-dual algorithms di er partly in the manner in which they achieve this reconciliation. The paper of Todd =-=[24]-=- is witness to the intensity of research in SDP interior-point methods: It describes twenty techniques for obtaining search directions for SDP, among the most notable being the following: 1) the AHO s... |

21 | A large-step infeasible-interior-point method for the P - matrix LCP
- Potra, Sheng
- 1994
(Show Context)
Citation Context ...he complexity estimates for interior-point methods applied to such problems depends on the parameter ,so that the complexity is not polynomial on the whole class of su cient matrices. Potra and Sheng =-=[19]-=- propose a large-step infeasible-interior-point method for solving P ( )-matrix linear complementarity problems with anumber of strong properties. The algorithm generates points in a large neighborhoo... |

18 | A unified analysis for a class of long-step primal-dual pathfollowing interior-point algorithms for semidefinite programming - Monteiro, Zhang - 1998 |

16 | R.: Superlinear convergence of interior-point algorithms for semidefinite programming
- Potra, Sheng
- 1998
(Show Context)
Citation Context ...k S k =n) =0: Assumption (B) and (C) are quite restrictive; similar conditions are not required for the superlinear convergence of interior-point methods for linear programming or QP. Potra and Sheng =-=[20]-=- proved superlinear convergence of the same algorithm under assumption (A) together with the following condition: (D) lim k!1 Xk S k = p X k S k =0, which is clearly weaker than (C). Of course both (C... |

15 | Interior methods for constrained optimization, in Acta Numerica - Wright - 1992 |

14 | An O(pnL)-iteration homogeneous and self-dual linear programming algorithm - Ye, Todd, et al. - 1994 |

13 | Interior point methods: current status and future directions
- Freund, Mizuno
- 2000
(Show Context)
Citation Context ...ming, semide nite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions. 1 Introduction In their survey article =-=[6]-=-, Freund and Mizuno wrote Interior-point methods in mathematical programming have been the largest and most dramatic area of research in optimization since the development of the simplex method::: Int... |

12 |
On the convergence of a class of infeasible-interior-point methods for the horizontal linear complementarity problem
- Zhang
- 1994
(Show Context)
Citation Context ...s, and 7sthey are more di cult to analyze. The rst global convergence result for such methods was obtained by Kojima, Megiddo and Mizuno, while the rst polynomial complexity result was given by Zhang =-=[32]-=-. The computational complexity of the infeasible-interior-point algorithms typically is worse than in the feasible case. An advantage is that these algorithms can solve problems for which no strictly ... |

9 |
Average performance of a self– dual interior–point algorithm for linear programming
- Anstreicher, Ji, et al.
- 1993
(Show Context)
Citation Context ...ity conditions Ax = b and A T + s = c are satis ed, the primal-dual step ( x; ; s) is obtained from following system: " 0 A 0 A T 0 I 0 S X #" x s # =, 5 " 0 0 XSe, e + r # ; (2.10)swhere = x T s=n,2 =-=[0; 1]-=-, and r is a perturbation term, possibly chosen to incorporate higher-order information about the system (2.8), or additional terms to improve proximity to the central path. Using the general step (2.... |

7 | A quadratically convergent O(pnL)-iteration algorithm for linear programming - Ye, Guler, et al. - 1993 |

3 | Recent developments in interior-point methods
- WRIGHT
- 1999
(Show Context)
Citation Context ...st regions. For references to work mentioned in the previous paragraph, and for many other results discussed but not cited in this paper, please see the bibliography of the technical report by Wright =-=[28]-=-. A great deal of literature is available to the reader interested in learning more about interior-point methods. Anumber of recent books [27], [29], [23] give overviews of the area, from rst principl... |

2 | Linear programming, in: G.L - Goldfarb, Todd - 1989 |

1 |
programming in O([n 3 =ln n]L) operations, CORE Discussion Paper 9746, Universite Catholique de Louvain
- Anstreicher, Linear
- 1999
(Show Context)
Citation Context ...ldfarb and Todd [8] provide a good reference for these complexity results. By using fast matrix multiplication techniques, the complexity estimates can be reduced further. Quite recently, Anstreicher =-=[2]-=- proposed an interior-point method, combining partial updating with a preconditioned gradient method, that has an overall complexity of O(n3 = log n) bit operations. The paper [2] contains references ... |

1 |
On the local convergence of asymmetric predictor-corrector method for semidefinite programming
- Ji, Potra, et al.
- 1997
(Show Context)
Citation Context ...al complexity bounds have yet been found. It appears that the use of the AHO direction in the corrector step has a strong e ect on centering. This property is exploited in a recent paper of Ji et al. =-=[10]-=- who proved that the Mizuno-Todd-Ye algorithm, based on the MZ-family is superlinear under assumptions (A) and (D). They also showed that under assumptions (A) and (B) the algorithm has Q-order 1.5 if... |

1 |
A uni ed analysis for a class of long-step primal–dual path-following interior-point algorithms for semide nite programming
- Monteiro, Zhang
- 1998
(Show Context)
Citation Context ...n n : (X; S) =0; (4.25) 12sPrimal-dual methods are then derived as perturbed Newton's methods applied to (4.24a), (4.24b), (4.25). Examples of symmetrizations (4.25) include the Monteiro-Zhang family =-=[15]-=-, in which where (X; S) =HP (XS); HP (M) = 1 2 PMP,1 +(PMP ,1 ) T ; (with a given a nonsingular matrix P2 IR n n ) is the symmetrization operator of Zhang. The search directions 1), 2), 3) mentioned a... |

1 |
Semide nite progamming
- Vandenberghe, Boyd
- 1996
(Show Context)
Citation Context ...e semide nite matrices. SDP is a broad paradigm; it includes as special cases linear programming, (linearly constrained) QP, quadratically constrained QP and other optimization problems (see [16] and =-=[25]-=-). Semide nite programming has numerous applications in such diverse areas as optimal control, combinatorial optimization, structural optimization, pattern recognition, trace factor analysis in statis... |

1 | programming in O([n = ln n]L) operations. CORE Discussion Paper 9746, Universit'e Catholique de Louvain - Linear - 1999 |

1 | Semidefinite progamming - Vandenberghe, Boyd - 1996 |

1 | programming in O([n3= ln n]L) operations. CORE Discussion Paper 9746, Universit'e Catholique de Louvain - Linear - 1999 |