A nonmonotone strategy for solving nonlinear systems of equations is introduced. The idea consists of combining eecient local methods with an algorithm that reduces monotonically the squared norm of the system in a proper way. The local methods used are Newton's method and two quasi-Newton algorithms. Global iterations are based on recently introduced box-constrained minimization algorithms. We...

In this paper, we consider the problem of efficiently computing the eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact formulation. In addition, we produce a compact formula for quasi-Newton matrices generated by any member of the Broyden convex class of updates. Our proposed method makes use of efficient updates to the QR factorization that substantially reduces the cos...

This paper reports theoretical and empirical investigations on the use of quasi-Newton methods to minimize the Optimal Bellman Residual (OBR) of zero-sum two-player Markov Games. First, it reveals that state-of-the-art algorithms can be derived by the direct application of Newton’s method to different norms of the OBR. More precisely, when applied to the norm of the OBR, Newton’s method results...

A derivative-free Quasi-Newton (DFQN) method previously published [J. Greenstadt, Math. Comp., v. 26, 1972, pp. 145—166] has been revised and simplified. The main modification has the effect of keeping all the successive approximants to the Hessian matrix positive-definite. This, coupled with some improvements in the line search, has enhanced the performance of the method considerably. The resu...

We first recall some properties of infinite tridiagonal matrices considered as matrix transformations in sequence spaces of the forms sξ , sξ , s (c) ξ , or lp(ξ). Then, we give some results on the finite section method for approximating a solution of an infinite linear system. Finally, using a quasi-Newton method, we construct a sequence that converges fast to a solution of an infinite linear ...

A classical model of Newton iterations which takes into account some error terms is given by the quasi-Newton method, which assumes perturbed Jacobians at each step. Its high convergence orders were characterized by Dennis and Moré [Math. Comp. 28 (1974), 549–560]. The inexact Newton method constitutes another such model, since it assumes that at each step the linear systems are only approximat...

We describe a parallel method for unconstrained optimization based on the quasi-Newton descent method of Broyden, Fletcher, Goldfarb, and Shanno. Our algorithm is suitable for both single-instruction and multiple-instruction parallel architectures and has only linear memory requirements in the number of parameters used to ®t the data. We also present the results of numerical testing on both sin...

A new method for the solution of minimization problems with simple bounds is presented. Global convergence of a general scheme requiring the solution of a single linear system at each iteration is proved and a superlinear convergence rate is established without requiring the strict complementarity assumption. The theory presented covers Newton and Quasi-Newton methods, allows rapid changes in t...

In this paper, non-monotone line search procedure is studied, which is combined with the non-quasi-Newton family. Under the uniformly convexity assumption on objective function, the global and superlinear convergence of the non-quasi-Newton family with the proposed nonmonotone line search is proved under suitable conditions.

This paper proposes a new predictor-corrector interior-point method for a class of semidefinite programs, which numerically traces the central trajectory in a space of Lagrange multipliers. The distinguished features of the method are full use of the BFGS quasi-Newton method in the corrector procedure, and an application of the conjugate gradient method with an effective preconditioning matrix ...