## An Interior-Point Method for Semidefinite Programming (2005)

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Citations: | 205 - 18 self |

### BibTeX

@MISC{Helmberg05aninterior-point,

author = {Christoph Helmberg and Franz Rendl and Robert J. Vanderbei and Henry Wolkowicz},

title = {An Interior-Point Method for Semidefinite Programming},

year = {2005}

}

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### Abstract

We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as max-cut. Other applications include max-min eigenvalue problems and relaxations for the stable set problem.

### Citations

477 | Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results
- Alizadeh, Haeberly, et al.
- 1994
(Show Context)
Citation Context ...me computational experiments indicating that the approach is also highly efficient in practice. We close this section by describing research related to our work. Alizadeh, Haeberly, Jarre and Overton =-=[2, 3, 4, 14, 15, 21, 22]-=- consider a problem similar to ours. Algorithmically, these authors use mostly interior point based techniques to solve the problem. Alizadeh proposes a potential reduction method and shows a polynomi... |

339 | On the Shannon capacity of a graph - Lovász - 1979 |

251 |
Interior-point polynomial methods in convex programming
- Nesterov, Nemirovsky
- 1994
(Show Context)
Citation Context ...uthors present several interior point approaches to this type of semidefinite program. Finally, a general framework for interior point methods applied to convex programs can be found in the monograph =-=[20]-=-. 1.1 Preliminaries We first collect some preliminary results and notation. We work mainly in the space M n of symmetric n \Theta n matrices, endowed with inner product hU; V i := tr(UV T ): The curly... |

119 |
A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis
- Schramm, Zowe
- 1992
(Show Context)
Citation Context ...A T g = 0 The last equality follows from the fact that the columns of A are eigenvectors of C associated with the maximal eigenvalue. Table 4 shows the comparison between the bundle trust method, see =-=[26]-=-, and our interior-point method when the optimal eigenvalue is a singleton (k = 1). For these problems, the bundle trust method is three to four times faster (computing times are given for a Silicon G... |

86 | Primal-dual potential reduction method for problems involving matrix inequalities
- Vandenberghe, Boyd
- 1995
(Show Context)
Citation Context ...1, 22, 4] are not in the form above, but it is an easy exercise to transform them into our model. Vandenberghe and Boyd study primal-dual potential reduction algorithms for semidefinite programs, see =-=[27]-=-. In [16], the monotone linear complementarity problem for symmetric matrices is investigated. The authors present several interior point approaches to this type of semidefinite program. Finally, a ge... |

80 | Large-scale optimization of eigenvalues
- Overton
(Show Context)
Citation Context ...me computational experiments indicating that the approach is also highly efficient in practice. We close this section by describing research related to our work. Alizadeh, Haeberly, Jarre and Overton =-=[2, 3, 4, 14, 15, 21, 22]-=- consider a problem similar to ours. Algorithmically, these authors use mostly interior point based techniques to solve the problem. Alizadeh proposes a potential reduction method and shows a polynomi... |

73 | Finding a maximum cut of a planar graph in polynomial time - HADLOCK - 1975 |

72 |
On minimizing the maximum eigenvalue of a symmetric matrix
- Overton
- 1988
(Show Context)
Citation Context ...me computational experiments indicating that the approach is also highly efficient in practice. We close this section by describing research related to our work. Alizadeh, Haeberly, Jarre and Overton =-=[2, 3, 4, 14, 15, 21, 22]-=- consider a problem similar to ours. Algorithmically, these authors use mostly interior point based techniques to solve the problem. Alizadeh proposes a potential reduction method and shows a polynomi... |

70 |
878-approximation algorithms for MAX CUT and MAX 2SAT
- Goemans, Williamson
- 1994
(Show Context)
Citation Context ...4 e rank(X) = 1 Xs0: Dropping the rank condition we obtain a problem of the form (SDP) with no inequalities, a = 1 4 e and A(X) = diag(X). This relaxation of max-cut is well known and studied e.g. in =-=[9, 11, 24]-=-. Goemans and Williamson [11] have recently shown that the optimal value of this relaxation is at most 14% above the value of the maximum cut, provided As0, i.e. no negative edge weights exist. The va... |

63 |
Combinatorial Optimization with interior point methods and semidefinite matrices
- Alizadeh
- 1991
(Show Context)
Citation Context |

63 |
Interior point methods for the monotone linear complementarity problem in symmetric matrices
- Kojima, Shindoh, et al.
(Show Context)
Citation Context ... are not in the form above, but it is an easy exercise to transform them into our model. Vandenberghe and Boyd study primal-dual potential reduction algorithms for semidefinite programs, see [27]. In =-=[16]-=-, the monotone linear complementarity problem for symmetric matrices is investigated. The authors present several interior point approaches to this type of semidefinite program. Finally, a general fra... |

59 |
Eigenvalues and graph bisection: an average case analysis
- Boppana
- 1987
(Show Context)
Citation Context ...is relaxation was also studied in [10] where it was treated as a min-max eigenvalue problem using nonsmooth optimization techniques. A more theoretical investigation of this bound is given by Boppana =-=[6]-=-. 3.3 Maximum cliques in graphs Semidefinite programs are also used in conjunction with stable set and clique problems in graphs, see [18]. Suppose a graph G on n vertices is given by its edge set E. ... |

56 |
Laplacian eigenvalues and the maximum cut problem
- Delorme, Poljak
- 1993
(Show Context)
Citation Context ...4 e rank(X) = 1 Xs0: Dropping the rank condition we obtain a problem of the form (SDP) with no inequalities, a = 1 4 e and A(X) = diag(X). This relaxation of max-cut is well known and studied e.g. in =-=[9, 11, 24]-=-. Goemans and Williamson [11] have recently shown that the optimal value of this relaxation is at most 14% above the value of the maximum cut, provided As0, i.e. no negative edge weights exist. The va... |

47 |
An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices
- Jarre
- 1993
(Show Context)
Citation Context |

45 |
The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices
- Cullum, Donath, et al.
- 1975
(Show Context)
Citation Context ...envalue exceeds one. In fact, a singleton eigenvalue characterizes differentiability. Since the largest eigenvalue is a convex function, subgradient approaches can be used to solve (6.12) (see, e.g., =-=[8]-=-). More recently, it has been shown that Newton-based algorithms with local quadratic convergence exist (see, e.g., [23]) but the local convergence depends on correctly identifying the multiplicity of... |

42 |
Symmetric indefinite systems for interior point methods
- Vanderbei, Carpenter
(Show Context)
Citation Context ...(ZX) + t T (b \Gamma B(X)) 2(n +m) : (4.9) (Experience from linear programming indicates that this simple heuristic performs very well, even though it does not guarantee monotonic decrease in ��, =-=see [28]-=-.) We next attempt to find directions (\DeltaX; \Deltay; \Deltat; \DeltaZ ) such that the new point (X + \DeltaX; y + \Deltay; t + \Deltat; Z+ \DeltaZ ) lies on the central trajectory at this value of... |

39 | Nonpolyhendral relaxations of graph-bisection problems
- Poljak, Rendl
- 1995
(Show Context)
Citation Context ...4 e rank(X) = 1 Xs0: Dropping the rank condition we obtain a problem of the form (SDP) with no inequalities, a = 1 4 e and A(X) = diag(X). This relaxation of max-cut is well known and studied e.g. in =-=[9, 11, 24]-=-. Goemans and Williamson [11] have recently shown that the optimal value of this relaxation is at most 14% above the value of the maximum cut, provided As0, i.e. no negative edge weights exist. The va... |

35 | A computational study of graph partitioning
- Falkner, Rendl, et al.
(Show Context)
Citation Context ... XJ = 0 Xs0: Here J = ee T is the matrix of all ones. Note that the constraint tr JX = 0 is obtained by squaring the cardinality constraint: 0 = (e T x) 2 = tr JX: This relaxation was also studied in =-=[10]-=- where it was treated as a min-max eigenvalue problem using nonsmooth optimization techniques. A more theoretical investigation of this bound is given by Boppana [6]. 3.3 Maximum cliques in graphs Sem... |

27 | Second derivatives for optimizing eigenvalues of symmetric matrices - Overton, Womersley - 1993 |

23 | Some applications of optimization in matrix theory
- Wolkowicz
- 1981
(Show Context)
Citation Context ...n , we let Diag(x) denote the diagonal matrix in M n whose diagonal elements are obtained from x. 2 Duality The general duality theory for problems such as (SDP) has been thoroughly studied, see e.g. =-=[30]-=-. We derive the dual to (SDP) directly using Lagrangian methods. Indeed, let ! denote the optimal objective value for (SDP). Introducing Lagrange multipliers y 2 ! k and t 2 ! m + for the equality and... |

23 | The max cut problem in graphs not contractible to Ks - BARAHONA - 1983 |

21 |
On the convergence of an infeasible primal-dual interior-point method for convex programming
- ANSTREICHER, VIAL
- 1994
(Show Context)
Citation Context ...a descent direction with respect to an appropriately defined merit function. We measure the progress of the algorithm using the following merit function. (This type of merit function was also used in =-=[1].) f �� -=-(X; y; t; Z) = hZ; Xi \Gamma �� log det(XZ) + t T (b \Gamma B(X)) \Gamma ��e T log(t ffi (b \Gamma B(X))) + (5.1) 1 2 jjF p jj 2 + 1 2 jjF d jj 2 For feasible points the merit function is the ... |

17 | Some applications of optimization in matrix theory. Linear Algebra and its Applications - Wolkowicz - 1981 |

17 | Combinatorial optimization with semidefinite matrices - Alizadeh - 1992 |

16 |
The Max-cut Problem on Graphs Not Contractible to K 5
- Barahona
- 1983
(Show Context)
Citation Context ...dered a computational challenge to optimize an arbitrary linear function over this polytope for say n �� 40: (If the graph is planar, then the metric relaxation already provides the max-cut, see e=-=.g. [5]-=-.) 3.2 Graph Bisection Graph bisection is similar to the max-cut problem, but here we seek a partition (S; T ) of the node set V such that the two sets have prespecified cardinalities, say jSj = k and... |

15 |
Higher-order predictor-corrector interior point methods with application to quadratic objectives
- Carpenter, Lustig, et al.
- 1993
(Show Context)
Citation Context ...ection found contains the information of the primal and dual variables at an equal degree. Both linearizations are especially well suited for Mehrotra's LP predictor--corrector method as described in =-=[7]-=-. The latter two statements also apply to the linearization of (5.10) as discussed in [4]. An advantage of this linearization is that it preserves symmetry. Furthermore Alizadeh, Haeberly, and Overton... |

12 |
The metric polytope
- Laurent, Poljak
- 1992
(Show Context)
Citation Context ...gain a relaxation for max-cut. This relaxation is usually called the Metric Relaxation, because the polyhedron fX : B(X) \Gamma bs0; diag(X) = ag is often referred to as the metric polytope, see e.g. =-=[17]-=-. We point out that this LP has i n 2 j variables and roughly 2 3 n 3 (very sparse) constraints. This polyhedron turns out to be highly degenerate, so that it is still considered a computational chall... |

12 |
ASZ. On the Shannon capacity of a graph
- LOV
- 1979
(Show Context)
Citation Context ...re theoretical investigation of this bound is given by Boppana [6]. 3.3 Maximum cliques in graphs Semidefinite programs are also used in conjunction with stable set and clique problems in graphs, see =-=[18]-=-. Suppose a graph G on n vertices is given by its edge set E. Define E ij := e i e t j +e j e t i ; where e j is column j of the identity matrix I n of size n. As above, J = ee t is the matrix of all ... |

11 |
Interior-point methods via self-concordance or relative Lipschitz condition
- Jarre
- 1995
(Show Context)
Citation Context |

10 | Max-min eigenvalue problems, primal-dual interior point algorithms, and trust region subproblems
- RENDL, VANDERBEI, et al.
- 1993
(Show Context)
Citation Context ...iable is poor. This was confirmed by practical experiments. Analogously, the linearization of (5.7) leads to A(XA T (\Deltay)X) = A(��X +XCX \Gamma XA T (y)X) \Gamma ��F p This formulation is =-=used in [25]-=-. This time the step is mainly based on the primal variable. This linearization can be considered as a pure primal approach. The linearization of (5.8) is the choice of this paper. It is easy to see t... |

10 | Computational experience with a primal-dual interior-point method for smooth convex programming", Technical report, Departement d'Economie Commerciale et Industrielle, Universite de Geneve - Vial - 1992 |

9 | A new primal–dual interior-point method for semidefinite programming - Alizadeh, Haeberly, et al. - 1994 |

5 | Eigenvalues in combinatorial optimization, in “Combinatorial and graph-theoretic problems in linear algebra - MOHAR, POLJAK - 1993 |

4 |
derivatives for optimizing eigenvalues of symmetric matrices
- Second
- 1993
(Show Context)
Citation Context ... convex function, subgradient approaches can be used to solve (6.12) (see, e.g., [8]). More recently, it has been shown that Newton-based algorithms with local quadratic convergence exist (see, e.g., =-=[23]-=-) but the local convergence depends on correctly identifying the multiplicity of the largest eigenvalue. We present computational experiments showing that our interior-point method is indeed robust in... |

3 |
Eigenvalues in combinatorial optimization, in "Combinatorial and graph-theoretic problems in linear algebra
- MOHAR, POLJAK
- 1993
(Show Context)
Citation Context ...on. We tacitly assume that the graph in question is complete (if not, nonexisting edges can be given weight 0 to complete the graph). Mathematically, the problem can be formulated as follows (see e.g =-=[19]-=-). Let the graph be given by its weighted adjacency matrix A. Define the matrix L := Diag(Ae) \Gamma A, where e is the vector of all ones. (The matrix L is called the Laplacian matrix associated with ... |

1 | On the Convergence of an Infeasible Primal22 Dual Interior-Point Method for Convex Programming - ANSTREICER, VIAL - 1993 |

1 | Interior point approach for max-cut - BURKARD, HELMBERG, et al. - 1994 |