82 research outputs found
BFGS-like updates of constraint preconditioners for sequences of KKT linear systems in quadratic programming
We focus on efficient preconditioning techniques for sequences of KKT linear systems
arising from the interior point solution of large convex quadratic programming problems.
Constraint Preconditioners~(CPs), though very effective in accelerating Krylov methods
in the solution of KKT systems, have a very high computational cost in some instances,
because their factorization
may be the most time-consuming task at each interior point iteration.
We overcome this problem by computing the CP from scratch only at selected interior point
iterations and by updating the last computed CP at the remaining iterations, via suitable
low-rank modifications based on a BFGS-like formula.
This work extends the limited-memory preconditioners for symmetric positive definite
matrices proposed by Gratton, Sartenaer and Tshimanga in [SIAM J. Optim. 2011; 21(3):912--935,
by exploiting specific features of KKT systems and CPs.
We prove that the updated preconditioners
still belong to the class of exact CPs, thus allowing the use of the conjugate gradient
method. Furthermore, they have the property of increasing the number of unit
eigenvalues of the preconditioned matrix as compared to generally used CPs.
Numerical experiments are reported, which show the effectiveness of our updating
technique when the cost for the factorization of the CP is high
Implementing a smooth exact penalty function for equality-constrained nonlinear optimization
We develop a general equality-constrained nonlinear optimization algorithm
based on a smooth penalty function proposed by Fletcher (1970). Although it was
historically considered to be computationally prohibitive in practice, we
demonstrate that the computational kernels required are no more expensive than
other widely accepted methods for nonlinear optimization. The main kernel
required to evaluate the penalty function and its derivatives is solving a
structured linear system. We show how to solve this system efficiently by
storing a single factorization each iteration when the matrices are available
explicitly. We further show how to adapt the penalty function to the class of
factorization-free algorithms by solving the linear system iteratively. The
penalty function therefore has promise when the linear system can be solved
efficiently, e.g., for PDE-constrained optimization problems where efficient
preconditioners exist. We discuss extensions including handling simple
constraints explicitly, regularizing the penalty function, and inexact
evaluation of the penalty function and its gradients. We demonstrate the merits
of the approach and its various features on some nonlinear programs from a
standard test set, and some PDE-constrained optimization problems
The antitriangular factorisation of saddle point matrices
Mastronardi and Van Dooren [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 173--196] recently introduced the block antitriangular (``Batman'') decomposition for symmetric indefinite matrices. Here we show the simplification of this factorization for saddle point matrices and demonstrate how it represents the common nullspace method. We show that rank-1 updates to the saddle point matrix can be easily incorporated into the factorization and give bounds on the eigenvalues of matrices important in saddle point theory. We show the relation of this factorization to constraint preconditioning and how it transforms but preserves the structure of block diagonal and block triangular preconditioners
Micro- and macro-block factorizations for regularized saddle point systems
We present unique and existing micro-block and induced macro-block Crout-based factorizations for matrices from regularized saddle-point problems with semi-positive de¿nite regularization block. For the classical case of saddle-point problems we show that the induced macro-block factorizations mostly reduces to the factorization presented in [24]. The presented factorization can be used as a direct solution algorithm for regularized saddle-point problems as well as it can be used a basis for the construction of preconditioners
An interior-point trust-region-based method for large-scale non-negative regularization
Abstract We present a new method for solving large-scale quadratic problems with quadratic and nonnegativity constraints. Such problems arise for example in the regularization of ill-posed problems in image restoration where, in addition, some of the matrices involved are very ill-conditioned. The new method uses recently developed techniques for the large-scale trust-region subproblem
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