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 $(A+\Delta_k) x_k=b_k$, where $A$ is symmetric positive semidefinite and $\Delta_k$ 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 … Read more

Efficient preconditioner updates for shifted linear systems

We present a new technique for building effective and low cost preconditioners for sequences of shifted linear systems (A+aI)x=b, where A is symmetric positive definite and a>0. This technique updates a preconditioner for A, available in the form of an LDL’ factorization, by modifying only the nonzero entries of the L factor in such a … Read more

Starting-Point Strategies for an Infeasible Potential Reduction Method

We present two strategies for choosing a “hot” starting-point in the context of an infeasible Potential Reduction (PR) method for convex Quadratic Programming. The basic idea of both strategies is to select a preliminary point and to suitably scale it in order to obtain a starting point such that its nonnegative entries are sufficiently bounded … Read more

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

The solution of KKT systems is ubiquitous in optimization methods and often dominates the computation time, especially when large-scale problems are considered. Thus, the effective implementation of such methods is highly dependent on the availability of effective linear algebra algorithms and software, that are able, in turn, to take into account specific needs of optimization. … Read more