Krylov Subspace Methods for Saddle Point Problems with Indefinite Preconditioning

Computational Optimization and Applications

Zilli

2004

120

128

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.

Section: Iterative Methodsmentioning

confidence: 95%

Section: Preconditionersmentioning

confidence: 99%

Section: Preconditioner 2 (Ne)mentioning

confidence: 99%

See 1 more Smart Citation

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Computational Optimization and Applications

Zilli

2004

120

128

“…Following the theory developed by Rozlozník and Simoncini [18], in [10] we studied the behaviour of the preconditioned conjugate gradient method on the indefinite system (8) preconditioned with (10). We can apply PCG method to an indefinite system because the following properties are satisfied.…”

Section: Convergence Of the Pcg Methodsmentioning

confidence: 99%

Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization

Al-Jeiroudi

2008

J Optim Theory Appl

In this paper we present the convergence analysis of the inexact infeasible path-following (IIPF) interior point algorithm. In this algorithm the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix.The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of interior point method for this specific inexact case. We present the convergence analysis of the inexact infeasible path-following (IIPF) algorithm, prove the global convergence of this method and provide complexity analysis.

“…The use of iterative methods such as conjugate gradients for indefinite systems may require some extra safeguards [18]. The subject has attracted a lot of attention in recent years [29,32]. It requires care to be taken about preconditioners [21,7] as well and/or special versions of the projected conjugate gradient method to be employed [24,15].…”

Section: From Augmented System To Normal Equationmentioning

confidence: 99%

Matrix-free interior point method

2010

Comput Optim Appl

In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach.