Computers: their operation and applications by Edmund Callis Berkeley

By Edmund Callis Berkeley

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When /~ < A then the optimal value of ,~* is ~* = - a l > 0 (since H is nonsingular for the case under consideration). Now (H + ~*I) is only positive semidefinite. , o. One could obtain a (non-optimal) direction, say p($*), from (19) 46 by setting this the first term to zero, giving p()r = - Q n - m .. Q,-mg (22) O-n - - 0-1 and again this is a direction of descent. 3 However, by assumption for the case under consideration, the second condition of (5) implies that the optimal solution w(A*) = z* - x of the model must satisfy IIw(M)H2 = A ( S is nonsingular so A* = - a ~ r 0).

See also Zhang and Tewarson [1987] for more recent developments. Section 5: The relationship between the BFGS (B-Update) and CG methods, oll which many linfited storage variants are founded, was developed by Nazareth [19765] and extended by Buckley [1978]. 40 CHAPTER THE MODEL-BASED 1 3 NEWTON PERSPECTIVE Introduction We again consider the general unconstrained minimization problem of Chapter 2, namely, (1) nfinimizexe~ f(x), where f : ~'~ --~ ~1 is a smooth nonlinear function. The classical method of Newton replaces the function at an approximate solution x by a quadratic function, which is derived, in turn, from a truncation of the Taylor expansion at x.

Thus, whenever/~ =/~T/~ > 0, the model-based Newton (unconstrained, positive-definite Hessian) approach and metric-based Cauchy approach are mathematically equivalent; we shall adopt the view that that they are two different ways of describing the same conceptual method, which we shall henceforth identify by the label: metric-based. , they may have different numerical properties and offer rather different opportunities for modification to enhance numerical stability. 2 and a variant of Procedure QN/B of Chapter 1, Section 8, which develops approximations Wj to the inverse Hessian (instead of Mj) and substitutes an exact line search for the formula for the step-length a~, so the modified QN/B procedure can be applied to an arbitrary smooth function.

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