A Regularized Interior-Point Method for Constrained Linear Least Squares

TL;DR: An infeasible interior-point algorithm for constrained linear least-squares problems based on the primal-dual regularization of convex programs of Friedlander and Orban is proposed and conditions under which a solution of the original, constrained least-Squares problem is recovered are established.

Abstract: RESUME : Nous proposons une methode de points interieurs non realisable pour le probleme aux moindres carres lineaire avec contraintes basee sur la regularisation primale-duale de problemes quadratiques convexes de Friedlander et Orban (2012). A chaque iteration, la methode effectue une factorisation LDLT creuse d’une matrice symetrique et quasi definie. Cette matrice est uniformement bornee et non singuliere. Nous etablissons des conditions sous lesquelles la methode produit une solution du probleme original. La regularisation nous permet d’eliminer l’hypothese que les gradients actifs sont lineairement independants. Bien que l’implementation proposee ici repose sur une factorisation, elle ouvre la voie a une implementation iterative dans laquelle on resout un probleme aux moindres carres regularise sans contraintes de facon inexacte a chaque iteration. Nous illustrons notre approche sur plusieurs applications qui mettent en evidence ses avantages.----------ABSTRACT : We propose an infeasible interior-point algorithm for constrained linear least-squares problems based on the primal-dual regularization of convex programs of Friedlander and Orban (2012). At each iteration, the sparse LDLT factorization of a symmetric quasi-definite matrix is computed. This coefficient matrix is shown to be uniformly bounded and nonsingular. We establish conditions under which a solution of the original problem is recovered. The regularization allows us to dispense with the assumption that the active gradients are linearly independent. Although the implementation described here is factorization based, it paves the way for a matrix-free implementation in which a regularized unconstrained linear least-squares problem is solved at each iteration. We report on computational experience and illustrate the potential advantages of our approach.