A new stopping criterion for Krylov solvers applied in Interior Point Methods

A surprising result is presented in this paper with possible far reaching consequences for any optimization technique which relies on Krylov subspace methods employed to solve the underlying linear equation systems. In this paper the advantages of the new technique are illustrated in the context of Interior Point Methods (IPMs). When an iterative method is applied to solve the linear equation system in IPMs, the attention is usually placed on accelerating their convergence by designing appropriate preconditioners, but the linear solver is applied as a black box solver with a standard termination criterion which asks for a sufficient reduction of the residual in the linear system. Such an approach often leads to an unnecessary 'oversolving' of linear equations. In this paper a new specialized termination criterion for Krylov methods used in IPMs is designed. It is derived from a deep understanding of IPM needs and is demonstrated to preserve the polynomial worst-case complexity of these methods. The new criterion has been adapted to the Conjugate Gradient (CG) and to the Minimum Residual method (MINRES) applied in the IPM context. The new criterion has been tested on a set of linear and quadratic optimization problems including compressed sensing, image processing and instances with partial differential equation constraints. Evidence gathered from these computational experiments shows that the new technique delivers significant improvements in terms of inner (linear) iterations and those translate into significant savings of the IPM solution time.