Showing papers by "John E. Dennis published in 1981"
TL;DR: NL2SOL is a modular program for solving nonlinear least-squares problems that incorporate a number of novel features and maintains a secant approximation S to the second-order part of the least-Squares Hessian and adaptively decides when to use this approximation.
Abstract: NL2SOL is a modular program for solving nonlinear least-squares problems that incorporate a number of novel features. It maintains a secant approximation S to the second-order part of the least-squares Hessian and adaptively decides when to use this approximation. S is "sized" before updating, something which is similar to Oren-Luenberger scaling. The step choice algorithm is based on minimizing a local quadratic model of the sum of squares function constrained to an elliptical trust region centered at the current approximate minimizer. This is accomplished using ideas discussed by More'', together with a special module for assessing the quality of the step thus computed. These and other ideas behind NL2SOL are discussed and its evolution and current implemetation are also described briefly.
865 citations
TL;DR: The algorithm amounts to a variation on Newton's method in which part of the Hessian matrix is computed exactly and part is approximated by a secant (quasi-Newton) updating method to promote convergence from poor starting guesses.
Abstract: Reference [ 1] explains the algorithm realized by NL2SOL in detail. The algorithm amounts to a variation on Newton's method in which part of the Hessian matrix is computed exactly and part is approximated by a secant (quasi-Newton) updating method. Once the iterates come sufficiently close to a local solution, they usually converge quite rapidly. To promote convergence from poor starting guesses, NL2SOL uses a model/trust-region technique along with an adaptive
403 citations
TL;DR: In this paper, a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means is presented, which can be viewed as generalizations of those given by Broyden-Dennis-More [J. Inst. Math. Appl. Comp., 28 (1974), pp. 549-560].
Abstract: The purpose of this paper is to present a convergence analysis of least change secant methods in which part of the derivative matrix being approximated is computed by other means. The theorems and proofs given here can be viewed as generalizations of those given by Broyden–Dennis–More [J. Inst. Math. Appl. 12 (1973), pp. 223–246] and by Dennis–More [Math. Comp., 28 (1974), pp. 549–560]. The analysis is done in the orthogonal projection setting of Dennis–Schnabel [SIAM Rev., 21(1980), pp. 443–459] and many readers might feel that it is easier to understand. The theorems here readily imply local and q-superlinear convergence of all the standard methods in addition to proving these results for the first time for the sparse symmetric method of Marwil and Toint and the nonlinear least-squares method of Dennis–Gay–Welsch.
125 citations