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18
A Spectral Bundle Method with Bounds
 MATHEMATICAL PROGRAMMING
, 1999
"... Semidefinite relaxations of quadratic 01 programming or graph partitioning problems are well known to be of high quality. However, solving them by primaldual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can sol ..."
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Cited by 33 (2 self)
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Semidefinite relaxations of quadratic 01 programming or graph partitioning problems are well known to be of high quality. However, solving them by primaldual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can solve quite efficiently large structured equalityconstrained semidefinite programs if the trace of the primal matrix variable is fixed, as happens in many applications. We extend the method so that it can handle inequality constraints without seriously increasing computation time. Encouraging preliminary computational results are reported.
On Copositive Programming and Standard Quadratic Optimization Problems
 Journal of Global Optimization
, 2000
"... A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the posi ..."
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Cited by 22 (5 self)
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A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FF^T where F is some nonnegative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a nonnegative quadratic form on the positive orthant). This conic formulation allows us to employ primaldual affinescaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primalfeasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primaldual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.
Solving Some Large Scale Semidefinite Programs Via the Conjugate Residual Method
, 2000
"... Most current implementations of interiorpoint methods for semidefinite programming use a direct method to solve the Schur complement equation (SCE) M y = h in computing the search direction. When the number of constraints is large, the problem of having insufficient memory to store M can be avoided ..."
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Cited by 21 (10 self)
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Most current implementations of interiorpoint methods for semidefinite programming use a direct method to solve the Schur complement equation (SCE) M y = h in computing the search direction. When the number of constraints is large, the problem of having insufficient memory to store M can be avoided if an iterative method is used instead. Numerical experiments have shown that the conjugate residual (CR) method typically takes a huge number of steps to generate a high accuracy solution. On the other hand, it is difficult to incorporate traditional preconditioners into the SCE, except for block diagonal preconditioners. We decompose the SCE into a 2 &times; 2 block system by decomposing y (similarly for h) into two orthogonal components with one lying in a certain subspace that is determined from the structure of M . Numerical experiments on semidefinite programming problems arising from Lovász function of graphs and MAXCUT problems show that high accuracy solutions can be obtained with moderate n...
A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations
 In The sharpest cut, MPS/SIAM Ser. Optim
, 2001
"... The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 01 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with re ..."
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Cited by 18 (2 self)
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The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 01 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 01 programming problems by oddcycle inequalities. We also consider separating oddcycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for maxcut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably.
A Semidefinite Programming Approach to the Quadratic Knapsack Problem
, 2000
"... In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibi ..."
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Cited by 14 (1 self)
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In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibilities to improve these basic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorial cuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational results on a small class of practical problems illustrate the quality of these relaxations and cutting planes.
A Note on the Calculation of StepLengths in InteriorPoint Methods for Semidefinite Programming
, 1999
"... . In each iteration of an interiorpoint method for semidefinite programming, the maximum steplength that can be taken by the iterate while maintaining the positive semidefiniteness constraint need to be estimated. In this note, we show how the maximum steplength can be estimated via the Lanczos i ..."
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Cited by 14 (4 self)
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. In each iteration of an interiorpoint method for semidefinite programming, the maximum steplength that can be taken by the iterate while maintaining the positive semidefiniteness constraint need to be estimated. In this note, we show how the maximum steplength can be estimated via the Lanczos iteration, a standard iterative method for estimating the extremal eigenvalues of a matrix. We also give a posteriori error bounds for the estimate. Key words. Semidefinite programming, steplength, Lanczos iteration. Department of Mathematics National University of Singapore 10 Kent Ridge Crescent, Singapore 119260, Singapore email: mattohkc@math.nus.edu.sg 1 Introduction Suppose X is an n \Theta n symmetric positive definite matrix that is the current primal iterate of an interior point method in semidefinite programming (SDP), and \DeltaX is the search direction to be taken. To define the next iterate, we need to estimate the maximum value the steplength ff can take while satisfyi...
Semidefinite Programs: New Search Directions, SmoothingType Methods, and Numerical Results
, 2001
"... Motivated by some results for linear programs and complementarity problems, this paper gives some new characterizations of the central path conditions for semidefinite programs. Exploiting these characterizations, some smoothingtype methods for the solution of semidefinite programs are derived. T ..."
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Cited by 12 (1 self)
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Motivated by some results for linear programs and complementarity problems, this paper gives some new characterizations of the central path conditions for semidefinite programs. Exploiting these characterizations, some smoothingtype methods for the solution of semidefinite programs are derived. The search directions generated by these methods are automatically symmetric, and the overall methods are shown to be globally and locally superlinearly convergent under suitable assumptions. Some numerical results are also included which indicate that the proposed methods are very promising and comparable to several interiorpoint methods. Moreover, the current method seems to be superior to the recently proposed smoothing method by Chen and Tseng [8].
Semidefinite programming for discrete optimization and matrix completion problems
 Discrete Appl. Math
, 2002
"... Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y ..."
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Cited by 11 (5 self)
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Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y
Semidefinite Programs and Association Schemes
, 1999
"... We consider semidenite programs, where all the matrices dening the problem commute. We show that in this case the semidenite program can be solved through an ordinary linear program. As an application, we consider the maxcut problem, where the underlying graph arises from an association scheme. ..."
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Cited by 7 (2 self)
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We consider semidenite programs, where all the matrices dening the problem commute. We show that in this case the semidenite program can be solved through an ordinary linear program. As an application, we consider the maxcut problem, where the underlying graph arises from an association scheme.
Semidefinite Programming
, 1999
"... Due to its many applications in control theory, robust optimization, combinatorial optimization and eigenvalue optimization, semidefinite programming had been in wide spread use even before the development of efficient algorithms brought it into the realm of tractability. Today it is one of the basi ..."
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Cited by 6 (0 self)
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Due to its many applications in control theory, robust optimization, combinatorial optimization and eigenvalue optimization, semidefinite programming had been in wide spread use even before the development of efficient algorithms brought it into the realm of tractability. Today it is one of the basic modeling and optimization tools along with linear and quadratic programming. Our survey is an introduction to semidefinite programming, its duality and complexity theory, its applications and algorithms.