Symmetric Quasidefinite Matrices

Vanderbei, Robert J.

doi:10.1137/0805005

Cited by 178 publications

(145 citation statements)

References 19 publications

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“…Quasidefinite matrix has the form [ −G A T A F ], where G and F are symmetric positive definite matrices and A has full rank. As shown in [28], quasidefinite matrices are strongly factorizable, i.e., a Cholesky-like factorization LDL T with a diagonal D exists for any symmetric row and column permutation of the quasidefinite matrix. The diagonal matrix D has n negative and m positive pivots.…”

Section: Introductionmentioning

confidence: 99%

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Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Gondzio

Zilli

2004

Computational Optimization and Applications

120

128

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Abstract. Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.

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Section: Introductionmentioning

confidence: 99%

“…An alternative is to transform the system to a quasidefinite one [28]. Quasidefinite matrix has the form [ −G A T A F ], where G and F are symmetric positive definite matrices and A has full rank.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Gondzio

Zilli

2004

Computational Optimization and Applications

120

128

View full text Add to dashboard Cite

show abstract

“…The pivoting strategies by Bunch and Parlett [11], Bunch and Kaufman [10] and Fletcher [29] are generally used in the latter factorization. When the matrices H and D are both positive definite, the KKT matrix is quasi-definite and a "sufficiently accurate" LBL T factorization with B diagonal can be computed [76]. Since the matrices to be factorized are often sparse, suitable reordering strategies are exploited to deal with the fill-in problem.…”

Section: Fundamental Issues In Solving the Kkt Systemmentioning

confidence: 99%

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

2008

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The solution of KKT systems is ubiquitous in optimization methods and often dominates the computation time, especially when large-scale problems are considered. Thus, the effective implementation of such methods is highly dependent on the availability of effective linear algebra algorithms and software, that are able, in turn, to take into account specific needs of optimization. In this paper we discuss the mutual impact of linear algebra and optimization, focusing on interior point methods and on the iterative solution of the KKT system. Three critical issues are addressed: preconditioning, termination control for the inner iterations, and inertia control.

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“…In our implementation of interior point method system (7) is regularized and then solved in two steps: (i) factorization of the form LDL T and (ii) backsolve to compute the directions (∆x, ∆y). Since the triangular decomposition exists for any symmetric row and column permutation of the quasi-definite matrix (Vanderbei 1995), the symbolic phase of the factorization (reordering for sparsity and preparing data structures for sparse triangular factor) can be done before the optimization starts.…”

Section: On Separate Processorsmentioning

confidence: 99%

“…to obtain a quasi-definite matrix (Vanderbei 1995) and allow more flexibility in the pivot ordering. The use of primal-dual regularization (8) guarantees the existence of a Cholesky-like LDL T factorization in which the diagonal D contains both positive and negative elements.…”

Section: On Separate Processorsmentioning

confidence: 99%

Parallel interior-point solver for structured linear programs

Gondzio¹,

Sarkissian²

2003

Mathematical Programming

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Many practical large-scale optimization problems are not only sparse, but also display some form of block-structure such as primal or dual block angular structure. Often these structures are nested: each block of the coarse top level structure is block-structured itself. Problems with these characteristics appear frequently in stochastic programming but also in other areas such as telecommunication network modelling.We present a linear algebra library tailored for problems with such structure that is used inside an interior point solver for convex quadratic programming problems. Due to its object-oriented design it can be used to exploit virtually any nested block structure arising in practical problems, eliminating the need for highly specialised linear algebra modules needing to be written for every type of problem separately. Through a careful implementation we achieve almost automatic parallelisation of the linear algebra.The efficiency of the approach is illustrated on several problems arising in the financial planning, namely in the asset and liability management. The problems are modelled as multistage decision processes and by nature lead to nested block-structured problems. By taking the variance of the random variables into account the problems become non-separable quadratic programs. A reformulation of the problem is proposed which reduces density of matrices involved and by these means significantly simplifies its solution by an interior point method. The object-oriented parallel solver achieves high efficiency by careful exploitation of the block sparsity of these problems. As a result a problem with over 50 million decision variables is solved in just over 2 hours on a parallel computer with 16 processors. The approach is by nature scalable and the parallel implementation achieves nearly perfect speedups on a range of problems.

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Symmetric Quasidefinite Matrices

Cited by 178 publications

References 19 publications

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

Parallel interior-point solver for structured linear programs

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