## Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization (1998)

Venue: | SIAM JOURNAL ON OPTIMIZATION |

Citations: | 114 - 11 self |

### BibTeX

@ARTICLE{Benson98solvinglarge-scale,

author = {Steven J. Benson and Yinyu Ye and Xiong Zhang},

title = {Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization},

journal = {SIAM JOURNAL ON OPTIMIZATION},

year = {1998},

volume = {10},

pages = {443--461}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the first computational results of interior-point algorithms for approximating the maximum cut semidefinite programs with dimension up-to 3000.

### Citations

937 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
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Citation Context ...were several theoretical results on the effectiveness of approximating combinatorial and nonconvex quadratic optimization problems by using semidefinite programming (see, e.g., Goemans and Williamson =-=[11]-=-, Nesterov [24], and Ye [33]). These results raise the hope that some hard optimization problems could be tackled by solving large-scale semidefinite relaxation programs. The positive semidefinite rel... |

770 | Semidefinite Programming
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Citation Context ...e solved in "polynomial time". There are actually several interior-point polynomial algorithms. One is the primal-scaling algorithm (Nesterov and Nemirovskii [25], Alizadeh [2], Vandenberghe=-= and Boyd [31]-=-, and Ye [34]), which is the analogue of the primal path-following and potential reduction algorithm for linear programming. This algorithm uses only X to generate the iterate direction. In other word... |

647 | A new polynomial-time algorithm for linear programming
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Citation Context ...te the reduction in the potential function from a current iterate (y k ; �� z k ) to the next, we will use a lemma from linear programming that can be found in [34] and is essentially do to Karmar=-=kar [15]-=-. Lemma 1 Let X 2 S n and kX \Gamma Ik 1 ! 1. Then ln det(X)str (X \Gamma I) \Gamma kX \Gamma Ik 2(1 \Gamma kX \Gamma Ik 1 ) where I denotes the identity matrix and k \Delta k denotes the Frobenius no... |

472 | Interior point methods in semidefinite programming with applications to combinatorial optimization
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Citation Context ...ll not converge to an optimal answer. There are also quite a few computational results and implementations of these interior algorithms, see Anstreicher and Fampa [4], Alizadeh, Haeberly, and Overton =-=[3]-=-, Fujisawa, Kojima and Nakata [8], Helmberg, Rendl, Vanderbei, and Wolkowicz [13], Karisch, Rendl, and Clausen [14], Vandenberghe and Boyd [31], Wolkowicz and Zhao [32], Zhao, Karisch, Rendl, and Wolk... |

336 |
On the Shannon capacity of a graph
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Citation Context ...ults raise the hope that some hard optimization problems could be tackled by solving large-scale semidefinite relaxation programs. The positive semidefinite relaxation was early considered by Lov'asz =-=[18]-=- and Shor [29], and the field had received further contributions by many other researchers (e.g., see Lov'asz and Shrijver [19], Alizadeh [2], Sherali and Adams [28], and references therein). The appr... |

217 |
Interior point algorithms: theory and analysis
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Citation Context ...polynomial time". There are actually several interior-point polynomial algorithms. One is the primal-scaling algorithm (Nesterov and Nemirovskii [25], Alizadeh [2], Vandenberghe and Boyd [31], an=-=d Ye [34]-=-), which is the analogue of the primal path-following and potential reduction algorithm for linear programming. This algorithm uses only X to generate the iterate direction. In other words, / X k+1 S ... |

202 | An interior-point method for semidefinite programming
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(Show Context)
Citation Context ...esults and implementations of these interior algorithms, see Anstreicher and Fampa [4], Alizadeh, Haeberly, and Overton [3], Fujisawa, Kojima and Nakata [8], Helmberg, Rendl, Vanderbei, and Wolkowicz =-=[13]-=-, Karisch, Rendl, and Clausen [14], Vandenberghe and Boyd [31], Wolkowicz and Zhao [32], Zhao, Karisch, Rendl, and Wolkowicz [35][36]. To the best of our knowledge, the largest problem that could be s... |

177 | Primal-dual interior-point methods for self-scaled cones
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Citation Context ...tions. The AHO direction [3] can be computed in 5nm 3 + n 2 m 2 + O(maxfm;ng 3 ) operations. The HRVW/KSH/M direction [13][16][21] uses 2nm 3 +n 2 m 2 +O(maxfm;ng 3 ) operations, and the NT direction =-=[26]-=- uses nm 3 + n 2 m 2 =2 + O(maxfm;ng 3 ) operations. The complexity of computing the matrix is a full order of magnitude higher than any other step of the algorithm. Fujisawa, Kojima and Nakata [8] ex... |

149 |
Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices
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Citation Context ...20] demonstrated an efficient implementation of several primal-dual step directions. The AHO direction [3] can be computed in 5nm 3 + n 2 m 2 + O(maxfm;ng 3 ) operations. The HRVW/KSH/M direction [13]=-=[16]-=-[21] uses 2nm 3 +n 2 m 2 +O(maxfm;ng 3 ) operations, and the NT direction [26] uses nm 3 + n 2 m 2 =2 + O(maxfm;ng 3 ) operations. The complexity of computing the matrix is a full order of magnitude h... |

138 | A spectral bundle method for semidefinite programming
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Citation Context ...kata [10] reported that they could solve a maximum cut semidefinite program with n = 1250, using a powerful work-station.) The practical winner of solving semidefinite programs was Helmberg and Rendl =-=[12]-=-, an implementation of a non-interior-point algorithm called the bundle method. They reported the solutions of a set of dual semidefinite programs with n up-to 3000. The bundle method enables them to ... |

115 |
Semidefinite relaxation and nonconvex quadratic optimization
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- 1998
(Show Context)
Citation Context ...eoretical results on the effectiveness of approximating combinatorial and nonconvex quadratic optimization problems by using semidefinite programming (see, e.g., Goemans and Williamson [11], Nesterov =-=[24]-=-, and Ye [33]). These results raise the hope that some hard optimization problems could be tackled by solving large-scale semidefinite relaxation programs. The positive semidefinite relaxation was ear... |

70 | Semidefinite programming relaxations for the quadratic assignment problem
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(Show Context)
Citation Context ..., Kojima and Nakata [8], Helmberg, Rendl, Vanderbei, and Wolkowicz [13], Karisch, Rendl, and Clausen [14], Vandenberghe and Boyd [31], Wolkowicz and Zhao [32], Zhao, Karisch, Rendl, and Wolkowicz [35]=-=[36]-=-. To the best of our knowledge, the largest problem that could be solved was at n = 900 from their reports. (After the initial version of this paper was submitted, one more implementation came out: Fu... |

68 | Approximating quadratic programming with bound and quadratic constraints
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- 1999
(Show Context)
Citation Context ...ults on the effectiveness of approximating combinatorial and nonconvex quadratic optimization problems by using semidefinite programming (see, e.g., Goemans and Williamson [11], Nesterov [24], and Ye =-=[33]-=-). These results raise the hope that some hard optimization problems could be tackled by solving large-scale semidefinite relaxation programs. The positive semidefinite relaxation was early considered... |

66 | Exploiting sparsity in primal-dual interiorpoint methods for semidefinite programming
- Fujisawa, Kojima, et al.
- 1997
(Show Context)
Citation Context ...wer. There are also quite a few computational results and implementations of these interior algorithms, see Anstreicher and Fampa [4], Alizadeh, Haeberly, and Overton [3], Fujisawa, Kojima and Nakata =-=[8]-=-, Helmberg, Rendl, Vanderbei, and Wolkowicz [13], Karisch, Rendl, and Clausen [14], Vandenberghe and Boyd [31], Wolkowicz and Zhao [32], Zhao, Karisch, Rendl, and Wolkowicz [35][36]. To the best of ou... |

63 |
Combinatorial optimization with interior point methods and semi-de nite matrices
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(Show Context)
Citation Context ...e semidefinite relaxation was early considered by Lov'asz [18] and Shor [29], and the field had received further contributions by many other researchers (e.g., see Lov'asz and Shrijver [19], Alizadeh =-=[2]-=-, Sherali and Adams [28], and references therein). The approximate solution to the quadratic optimization problem is obtained by solving a semidefinite relaxation, i.e., a semidefinite program (SDP) o... |

56 | An implementation of Karmarkar's algorithm for linear programming
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Citation Context ...imal-dual algorithms may not be utilized in our applications. The dual-scaling algorithm has been shown to perform equally well when only a lower precision answer is required, see, e.g., Adler et al. =-=[1]-=- and Vandenberghe and Boyd [31]. 2. In most combinatorial applications we need only a lower bound for the optimal objective value of (SDP). Solving (DSDP) alone would be sufficient to provide such a l... |

48 |
A unified analysis for a class of pathfollowing primal- dual interior-point algorithms for semidefinite programming, Math.program
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(Show Context)
Citation Context ...demonstrated an efficient implementation of several primal-dual step directions. The AHO direction [3] can be computed in 5nm 3 + n 2 m 2 + O(maxfm;ng 3 ) operations. The HRVW/KSH/M direction [13][16]=-=[21]-=- uses 2nm 3 +n 2 m 2 +O(maxfm;ng 3 ) operations, and the NT direction [26] uses nm 3 + n 2 m 2 =2 + O(maxfm;ng 3 ) operations. The complexity of computing the matrix is a full order of magnitude highe... |

38 |
A recipe for semidefinite relaxation for 0-1 quadratic programming
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Citation Context ...s that maximize the sum of the weighted edges connecting vertices in one set with vertices in the other. The positive semidefinite relaxation of the maximum cut problem can be expressed as (e.g., [11]=-=[27]-=-) Minimize C ffl X (MAX-CUT) Subject to diag(X) = e; Xs0: (18) The operator diag(\Delta) takes the diagonal of a matrix and makes it a vector. In other words, A i = e i e T i , i = 1; :::; n, where e ... |

35 | A computational study of graph partitioning
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(Show Context)
Citation Context ...is the dimension and the percentage of nonzero entries in the objective matrix, as well as the time (in seconds) and number of iterations required by the program. Most of the previous numerical tests =-=[7]-=-[17][32][35][36], were conducted on smaller problem data sets where the dimension n was only a few hundred or less, so that no available computation result could be compared to ours. After our results... |

32 | Semidefinite programming relaxation for nonconvex quadratic programs - Fujie, Kojima - 1997 |

31 |
On search directions in interior-point methods for semidefinite programming
- Todd
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(Show Context)
Citation Context ... iterate: / X k+1 S k+1 ! = F d (S k ); where F d is the dual algorithm iterative mapping. The third is the primal-dual scaling algorithm which uses both X and S to generate the new iterate (see Todd =-=[30]-=- and references therein): / X k+1 S k+1 ! = F pd (X k ; S k ); where F pd is the primal-dual algorithm iterative mapping. All these algorithms generate primal and dual iterates simultaneously, and pos... |

30 | Numerical Evaluation of SDPA (SemiDefinite Programming Algorithm
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(Show Context)
Citation Context ...e largest problem that could be solved was at n = 900 from their reports. (After the initial version of this paper was submitted, one more implementation came out: Fujisawa, Fukuda, Kojima and Nakata =-=[10]-=- reported that they could solve a maximum cut semidefinite program with n = 1250, using a powerful work-station.) The practical winner of solving semidefinite programs was Helmberg and Rendl [12], an ... |

25 | Solving graph bisection problems with semidefinite programming
- Karisch, Rendl, et al.
(Show Context)
Citation Context ...e interior algorithms, see Anstreicher and Fampa [4], Alizadeh, Haeberly, and Overton [3], Fujisawa, Kojima and Nakata [8], Helmberg, Rendl, Vanderbei, and Wolkowicz [13], Karisch, Rendl, and Clausen =-=[14]-=-, Vandenberghe and Boyd [31], Wolkowicz and Zhao [32], Zhao, Karisch, Rendl, and Wolkowicz [35][36]. To the best of our knowledge, the largest problem that could be solved was at n = 900 from their re... |

20 |
An Introduction to Numerical Analysis, Second Edition
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Citation Context ...z k \Gamma b T y k 1 + jb T y k j : We let the initial point X 0 = I and y 0 i = C ii \Gamma X j 6=i jC ij j \Gamma 1; i = 1; :::; n; which by Gerschgorin's Theorem, guarantees S 0 �� 0 (see Atkin=-=son [5]-=-). This value generally provides a reasonable starting point. We have used the minimum degree ordering algorithm to reorder C. We have stopped the iteration process when the relative duality gap rgap ... |

20 | Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants. Optim. Methods Softw
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Citation Context ...emidefinite programming requires a significant amount of the time to compute the system of equations that determines the step direction. For arbitrary symmetric matrices A i , Monteiro and Zanj'acomo =-=[20]-=- demonstrated an efficient implementation of several primal-dual step directions. The AHO direction [3] can be computed in 5nm 3 + n 2 m 2 + O(maxfm;ng 3 ) operations. The HRVW/KSH/M direction [13][16... |

17 | Sparse numerical linear algebra: direct methods and preconditioning
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Citation Context ... structure to save computation time. The table also emphasizes the importance of a good ordering of the matrix in the beginning of the algorithm. The reordering of matrices has been studied for years =-=[6]-=-, but to illustrate its importance, we include a few figures. The following figures in Figure 1 show the structure of S and its Cholesky factor in the 800 \Theta 800 example G14. The objective matrix ... |

14 |
A long-step path following algorithm for semide nite programming problems
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- 1996
(Show Context)
Citation Context ... direction. In other words, / X k+1 S k+1 ! = F p (X k ); where F p is the primal algorithm iterative mapping. Another is the dual-scaling algorithm (Vandenberghe and Boyd [31], Anstreicher and Fampa =-=[4]-=-, and Ye [34]), which is the analogue of the dual path-following and potential reduction algorithm for linear programming. The dual-scaling algorithm uses only S to generate the new iterate: / X k+1 S... |

13 | Semidefinite programming for assignment and partitioning problems
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(Show Context)
Citation Context ...sawa, Kojima and Nakata [8], Helmberg, Rendl, Vanderbei, and Wolkowicz [13], Karisch, Rendl, and Clausen [14], Vandenberghe and Boyd [31], Wolkowicz and Zhao [32], Zhao, Karisch, Rendl, and Wolkowicz =-=[35]-=-[36]. To the best of our knowledge, the largest problem that could be solved was at n = 900 from their reports. (After the initial version of this paper was submitted, one more implementation came out... |

8 |
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Citation Context ...on was early considered by Lov'asz [18] and Shor [29], and the field had received further contributions by many other researchers (e.g., see Lov'asz and Shrijver [19], Alizadeh [2], Sherali and Adams =-=[28]-=-, and references therein). The approximate solution to the quadratic optimization problem is obtained by solving a semidefinite relaxation, i.e., a semidefinite program (SDP) of the form Minimize C ff... |

7 |
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Citation Context ...ms. The positive semidefinite relaxation was early considered by Lov'asz [18] and Shor [29], and the field had received further contributions by many other researchers (e.g., see Lov'asz and Shrijver =-=[19]-=-, Alizadeh [2], Sherali and Adams [28], and references therein). The approximate solution to the quadratic optimization problem is obtained by solving a semidefinite relaxation, i.e., a semidefinite p... |

6 |
R.: On solving large scale semidefinite programming problems: a case study of quadratic assigment problem
- Lin, Saigal
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(Show Context)
Citation Context ...the dimension and the percentage of nonzero entries in the objective matrix, as well as the time (in seconds) and number of iterations required by the program. Most of the previous numerical tests [7]=-=[17]-=-[32][35][36], were conducted on smaller problem data sets where the dimension n was only a few hundred or less, so that no available computation result could be compared to ours. After our results rep... |

6 | Affine scaling algorithm fails for semidefinite programming
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- 1996
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Citation Context ...Other scaling algorithms have been proposed in the past. For example, an SDP equivalent of Dikin's affine-scaling algorithm could be very fast. However this algorithm may not even converge. Muramatsu =-=[22]-=- and Muramatsu and Vanderbei [23] showed an example in which these affine scaling algorithms will not converge to an optimal answer. There are also quite a few computational results and implementation... |

2 |
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Citation Context ... hope that some hard optimization problems could be tackled by solving large-scale semidefinite relaxation programs. The positive semidefinite relaxation was early considered by Lov'asz [18] and Shor =-=[29]-=-, and the field had received further contributions by many other researchers (e.g., see Lov'asz and Shrijver [19], Alizadeh [2], Sherali and Adams [28], and references therein). The approximate soluti... |

1 |
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Citation Context ... Alizadeh, Haeberly, and Overton [3], Fujisawa, Kojima and Nakata [8], Helmberg, Rendl, Vanderbei, and Wolkowicz [13], Karisch, Rendl, and Clausen [14], Vandenberghe and Boyd [31], Wolkowicz and Zhao =-=[32]-=-, Zhao, Karisch, Rendl, and Wolkowicz [35][36]. To the best of our knowledge, the largest problem that could be solved was at n = 900 from their reports. (After the initial version of this paper was s... |