Results 1  10
of
17
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 ..."
Abstract

Cited by 32 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 24 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 22 (11 self)
 Add to MetaCart
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 × 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 ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
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 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 ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
. 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...
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 ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
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.
Some New Search Directions for PrimalDual Interior Point Methods in Semidefinite Programming
"... Search directions for primaldual pathfollowing methods for semidefinite programming (SDP) are proposed. These directions have the properties that (1) under certain nondegeneracy and strict complementarity assumptions, the Jacobian matrix of the associated symmetrized Newton equation has bounded co ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
Search directions for primaldual pathfollowing methods for semidefinite programming (SDP) are proposed. These directions have the properties that (1) under certain nondegeneracy and strict complementarity assumptions, the Jacobian matrix of the associated symmetrized Newton equation has bounded condition number along the central path in the limit as the barrier parameter tends to zero; (2) the Schur complement matrix of the symmetrized Newton equation is symmetric and the cost for computing this matrix is 2mn 3 + 0:5m 2 n 2 ops, where n and m are the dimension of the matrix and vector variables of the SDP, respectively. These two properties imply that a pathfollowing method using the proposed directions can achieve the high accuracy typically attained by methods employing the direction proposed by Alizadeh, Haeberly, and Overton (currently the best search direction in terms of accuracy), but each iteration requires at most half the amount of flops (to leading order).