A Preconditioning Framework for Sequences of Diagonally Modified Linear Systems Arising in Optimization

Abstract

We propose a framework for building preconditioners for sequences of linear systems of the form (A + ∆ k)x k = b k , where A is symmetric positive semidefinite and ∆ k is diagonal positive semidefinite. Such sequences arise in several optimization methods, e.g., in affine-scaling methods for bound-constrained convex quadratic programming and bound-constrained linear least squares, as well as in trust-region and overestimation methods for convex unconstrained optimization problems and nonlinear least squares. For all the matrices of a sequence, the preconditioners are obtained by updating any preconditioner for A available in the LDL T form. The preconditioners in the framework satisfy the natural requirement of being effective on slowly varying sequences; furthermore, under an additional property they are also able to cluster eigenvalues of the preconditioned matrix when some entries of ∆ k are sufficiently large. We present two low-cost preconditioners sharing the abovementioned properties and evaluate them on sequences of linear systems generated by the reflective Newton method applied to bound-constrained convex quadratic programming problems, and on sequences arising in solving nonlinear least-squares problems with the Regularized Euclidean Residual method. The results of the numerical experiments show the effectiveness of these preconditioners.

An adaptive truncation criterion, for linesearch-based truncated Newton methods in large scale nonconvex optimization

“…The latter case is particularly appealing to our purposes, since it allows to exploit the strong relation between BFGS and the conjugacy of search directions with respect to matrix A [26] . Indeed, the BFGS update (2.3) is explicitly given by 5) and recalling the expression of f ( x ), along with relation y k = As k , we can set 6) where the vectors { p 1 , . .…”

“…Similarly, matrix-free preconditioning for linear systems or sequences of linear systems is currently an appealing research topic, too (see e.g. [4,5,13,14] ).…”

Modificação no Cálculo do Parâmetro de Correção de Falhas na Diagonal da Fatoração Controlada de Cholesky aplicado ao Método de Pontos Interiores

“…, 15 representa o número de reinícios permitidos na FCC. Por otro lado, baseado em [1], existe a FCCβ que evita reinícios quando existe falhas na diagonal da FCC, veja [7]. O desempenho da FCCβé discrepante em relaçãoà FCC, pois evitar reinícios acarreta num precondicionador ineficiente.…”

“…Na abordagem 2 híbrida proposta em [2] as falhas que ocorrem durante a fatoração são corrigidas com um incremento exponencial e o cálculo dos elementos da matriz precondicionadoraé reiniciado. Por outro lado, baseado no trabalho de [1], existe o precondicionador FCCβ que evita reinícios, esta propostaé apresentada em [7]. Os resultados obtidos pela FCCβ não são competitivos com os obtidos pela construção original porqué o tempo computacional em alguns problemas cresce consideravelmente.…”

On Partial Cholesky Factorization and a Variant of Quasi-Newton Preconditioners for Symmetric Positive Definite Matrices

“…The numerical solution of linear algebraic systems with symmetric positive definite (SPD) matrix is required in a broad range of applications, see e.g., [1][2][3][4][5][6]. We consider the case where the linear systems are large and investigate their iterative solution by preconditioned Krylov subspace methods [4,7,8].…”