A preconditioning framework for sequences of diagonally modified linear systems arising in optimization

We propose a framework for building preconditioners for sequences of linear systems of the form (A+Δk)xk = bk, where A is symmetric positive semidefinite and Δk 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(Table Presented) 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 Δk are sufficiently large. We present two low-cost preconditioners sharing the above-mentioned 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. © 2012 Society for Industrial and Applied Mathematics.