In this work a Newton interior–point method for the solution of Karush– Kuhn–Tucker systems is presented. A crucial feature of this iterative method is the solution, at each iteration, of the inner subproblem. This subproblem is a linear–quadratic programming problem, that can solved approximately by an inner iterative method such as the Hestenes multipliers’ method. A deep analysis on the choi...

Newton's Method constitutes a nested iteration scheme with the Newton step as the outer iteration and a linear solver of the Jacobian system as the inner iteration. We examine the interaction between these two schemes and derive solution techniques for the linear system from the properties of the outer Newton iteration. Contrary to inexact Newton methods, our techniques do not rely on relaxed t...

Gaussian processes allow the treatment of non-linear non-parametric regression problems in a Bayesian framework. However the computational cost of training such a model with N examples scales as O(N3). Iterative methods for the solution of linear systems can bring this cost down to O(N2), which is still prohibitive for large data sets. We consider the use of 2-exact matrix-vector product algori...

We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimization subproblems. The solver constructs the primal and dual trial steps from the subspace generated by the generalized Arnoldi procedure used in flexible GMRES (FGMRES). This permits the use of a wide range of preconditioners for the primal-dual system. In contrast with FGMRES, the proposed method se...

in this paper, we represent an inexact inverse subspace iteration method for com- puting a few eigenpairs of the generalized eigenvalue problem ax = bx[q. ye and p. zhang, inexact inverse subspace iteration for generalized eigenvalue problems, linear algebra and its application, 434 (2011) 1697-1715 ]. in particular, the linear convergence property of the inverse subspace iteration is preserved.

The paper deals with three different Newton algorithms that have recently been worked out in the general frame of affine invariance. Of particular interest is their performance in the numerical solution of discretized boundary value problems (BVPs) for nonlinear partial differential equations (PDEs). Exact Newton methods, where the arising linear systems are solved by direct elimination, and in...

For the iterative solution of saddle point problems, a nonsymmetric preconditioner is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves soluti...

Variable projection solves structured optimization problems by completely minimizing over a subset of the variables while iterating remaining variables. Over past 30 years, technique has been widely used, with empirical and theoretical results demonstrating both greater efficacy stability compared to competing approaches. Classic examples have exploited closed-form projections smoothness object...