In this paper, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers, which in turn results in a primal-dual regularized interior point method. Application of this method gives rise to a … Read more
We address the iterative solution of symmetric KKT systems arising in the solution of convex quadratic programming problems. Two strictly related and well established formulations for such systems are studied with particular emphasis on the effect of preconditioning strategies on their relation. Constraint and augmented preconditioners are considered, and the choice of the augmentation Matrix … Read more
We consider symmetrized KKT systems arising in the solution of convex quadratic programming problems in standard form by Interior Point methods. Their coefficient matrices usually have 3×3 block structure and, under suitable conditions on both the quadratic programming problem and the solution, they are nonsingular in the limit. We present new spectral estimates for these … Read more
We propose a class of preconditioners for symmetric linear systems arising from numerical analysis and nonconvex optimization frameworks. Our preconditioners are specifically suited for large indefinite linear systems and may be obtained as by-product of Krylov-subspace solvers, as well as by applying L-BFGS updates. Moreover, our proposal is also suited for the solution of a … Read more
This work is concerned with the development and study of a class of limited memory preconditioners for the solution of sequences of linear systems. To this aim, we consider linear systems with the same symmetric positive definite matrix and multiple right-hand sides available in sequence. We first propose a general class of preconditioners, called Limited … Read more
We propose primal-dual path-following Mehrotra-type predictor-corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: $\min_{X} \{ \frac{1}{2} X\bullet {\cal Q}(X) + C\bullet X : {\cal A} (X) = b, X\succeq 0\}$, where ${\cal Q}$ is a self-adjoint positive semidefinite linear operator on ${\cal S}^n$, $b\in R^m$, and ${\cal A}$ is a … Read more
The search directions in an interior-point method for large scale semidefinite programming (SDP) can be computed by applying a Krylov iterative method to either the Schur complement equation (SCE) or the augmented equation. Both methods suffer from slow convergence as interior-point iterates approach optimality. Numerical experiments have shown that diagonally preconditioned conjugate residual method on … Read more