In this work we describe two sequential algorithms and their parallel counterparts for solving nonlinear systems, when the Jacobian matrix is symmetric and positive definite. This case appears frequently in unconstrained optimization problems. Both algorithms are based on Newton’s method. The first solves the inner iteration with Cholesky decomposition while the second is based on the inexact N...

A new algorithm for the solution of large-scale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact Levenberg-Marquardt-type algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be b...

A decomposition algorithm based on proximal bundle-type method with inexact data is presented for minimizing an unconstrained nonsmooth convex function f . At each iteration, only the approximate evaluation of f and its approximate subgradients are required which make the algorithm easier to implement. It is shown that every cluster of the sequence of iterates generated by the proposed algorith...

In this paper, we present a QP-free algorithm for nonlinear semidefinite programming. At each iteration, the search direction is yielded by solving two systems of linear equations with the same coefficient matrix; [Formula: see text] penalty function is used as merit function for line search, the step size is determined by Armijo type inexact line search. The global convergence of the proposed ...

In this paper we analyze the use of structured quasi-Newton formulae as preconditioners of iterative linear methods when the inexact-Newton approach is employed for solving nonlinear systems of equations. We prove that superlinear convergence and bounded work per iteration is obtained if the preconditioners satisfy a Dennis-Moré condition. We develop a theory of LeastChange Secant Update precon...

− A new inverse scattering method is assessed against some of the real input data measured by the Institut Fresnel, Marseille, France. The method is based on the application of an Inexact-Newton method to the Lippmann-Schwinger integral equation of the inverse scattering problem within the second-order Born approximation. The regularization properties of the approach are evaluated by considerin...

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 16 January 2020Accepted: 09 July 2021Published online: 25 October 2021Keywordsquadratic penalty method, composite nonconvex problem, iteration complexity, inexact proximal point first-order accelerated gradient minimax problemAMS Subject Headings47J22, 90C26, 90C30, 90C47...

We incorporate our recent preconditioning techniques into the classical inverse power (Rayleigh quotient) iteration for computing matrix eigenvectors. Every loop of this iteration essentially amounts to solving an ill conditioned linear system of equations. Due to our modification we solve a well conditioned linear system instead. We prove that this modification preserves local quadratic conver...

The purpose of this paper is two-fold: to analyze the behavior of inverse iteration for computing a single eigenvector of a complex square matrix and to review Jim Wilkinson’s contributions to the development of the method. In the process we derive several new results regarding the convergence of inverse iteration in exact arithmetic. In the case of normal matrices we show that residual norms d...

In this work, we consider a stochastic Nash game in which each player solves a parameterized stochastic optimization problem. In deterministic regimes, best-response schemes have been shown to be convergent under a suitable spectral property associated with the proximal best-response map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to s...