Unconstrained large-scale

R. S. Dembo and T. Steihaug, Math. Prog., 26, 190 (1983). Truncated-Newton Algorithms for Large-Scale Unconstrained Optimization. 

This chapter deals with the problem of finding the unconstrained minimum of a function P x) that involves the variables x e R with Wv 1. Section 3.4 showed that conjugate direction methods are useful in solving large-scale unconstrained minimization problems. The version that uses the Pollack-Ribiere and Fletcher-Reeves methods sequentially is often particularly effective. 

It is possible to optimize a large-scale unconstrained system only if the function can be reasonably approximated by a quadratic function. 

The class used in the BzzMath library for large-scale unconstrained minimization is... 

Chapter 4 has been devoted to large-scale unconstrained optimization problems, where problems related to the management of matrix sparsity and the ordering of rows and columns are broached. Hessian evaluation, Newton and inexact Newton methods are discussed. 

Nonlinear optimization is one of the crucial topics in the numerical treatment of chemical engineering problems. Numerical optimization deals with the problems of solving systems of nonlinear equations or minimizing nonlinear functionals (with respect to side conditions). In this article we present a new method for unconstrained minimization which is suitable as well in large scale as in bad conditioned problems. The method is based on a true multi-dimensional modeling of the objective function in each iteration step. The scheme allows the incorporation of more given or known information into the search than in common line search methods. 

The sloping solid line shows the reported temperature variation of T2 between 13 K and 17 K for unconstrained solid D2 with an x = 0.33 p-D2 fraction. The dashed curve shows the coefficient of self-diffusion (on the right hand scale) reported for liquid n-H2 at SVP. Liquid D2 diffusion must follow a similar curve. It is probable that the observed temperature variation of T2 for the narrow central DMR component in a-Si D,F (325) reflects the melting of bulk solid and diffusion in dense fluid D2 in microvoids. Either the presence of F produces unusually large voids (which does not seem likely) or else void surfaces are rendered less effective in controlling the relaxation properties of the contained D2 than was the case in a-Si D,H (circles). 

The magnitude of R in an unconstrained chain is uncorrelated with its direction, so that its value averaged over a large number of arbitrarily chosen conformations must be zero. Because thermal motion causes a real chain to change its local conformation every few picoseconds, for more familiar time scales it would therefore be perceived as a fuzzy ball with = 0 and a time-averaged mean square end-to-end distance given by Eq. (2) (r and im are uncorrdated so that [Pg.723]

---