Preconditioning Indefinite Systems in Interior-Point Methods for quadratic optimization

A new class of preconditioners is proposed for the iterative solution of symmetric indefinite systems arising from interior-point methods. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Now we introduce two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems. For convex optimization problems all the eigenvalues of this matrix are strictly positive. Meanwhile, we apply the new regularization techniques for symmetric indefinite systems to improve the stability of our iterative approach. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results provides the potential advantages of our approach when applied to the solution of very large quadratic problems. Keywords-Preconditioners; Interior-Point Methods; Indefinite Systems; Quadratic Problems