By J.L. Nazareth
ISBN-10: 3540576711
ISBN-13: 9783540576716
Computational unconstrained nonlinear optimization involves lifestyles from a research of the interaction among the metric-based (Cauchy) and model-based (Newton) issues of view. The motivating challenge is that of minimizing a convex quadratic functionality. This learn monograph finds for the 1st time the basic harmony of the topic. It explores the relationships among the most equipment, develops the Newton-Cauchy framework and issues out its wealthy wealth of algorithmic implications and easy conceptual equipment. The monograph additionally makes a valueable contribution to unifying the notation and terminology of the topic. it truly is addressed topractitioners, researchers, teachers, and scholars and offers an invaluable and fresh new point of view on computational nonlinear optimization.
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Additional resources for The Newton-Cauchy Framework: A Unified Approach to Unconstrained Nonlinear Minimization
Example text
37 g~. >0 S(x~ + ~d~). xj+l = xj + c~dj. Evaluate gj+l. sj = x j + l - - x j ; Yj = g j + l - - g j . i-1. ~--"5-Yj _ l ttj --1 (59) This is precisely the Hestenes-Stiefel conjugate gradient direction (1:35), but now it has been derived in a conceptually more general and satisfactol T way. 4) are identical for arbitrary smooth functions, which can be seen as follows: yf_ldj-1 = -gf_ldj_l, because g f dj_l -= O = g L ~ g ~ - , , because gT-'e~-~ = O. Thus (59) can be written in the equivalent Polak-Ribiere form @,gJ dj = - g j -4- g f _ l g j _ l d j - 1 9 [] (60) Under the foregoing assumptions, the Fletcher-Reeves choice, which we may also note is always nonnegative, is not equivalent to the other two.
However, should we seek to satisfy the equation IIw($*)]12 = A, the optimal solution would be dominated by the columns of Qm and we would not have a direction of descent. We must be content with the descent direction p($*) defined by (22). An alternative is to choose a large but finite value for A and proceed as in case c) (ii). Thus, in all four cases we have found a suitable search direction of descent. The heart of the procedure is an eigendecomposition of H and a zero-finding procedure to solve a nonlinear equation in one variable, namely, I l w ( ~ ) ] 1 2 - z~ = o, where w(A) is defined by (17).
Note that the objective function is nonnegative and replacing it by its square does not alter the optimal solution of (34)-(35). t. (R + uvT)s = (R + uvT)-T y. (36) This can be solved with the help of the standard Lagrangian optimality conditions. However, the special form of (36) also makes the problem amenable to solution by the following three-stage procedure that is especially tailored to it: 1. t. where the third constraint follows directly from the previous two. 2. t. (R + ~vT)T~ = V- (42) Denote the optimal solution of (41)-(42) by u(w) and v(w) and the corresponding optimal objective value by [u(~) T~(~)][~(~)~(~)].
The Newton-Cauchy Framework: A Unified Approach to Unconstrained Nonlinear Minimization by J.L. Nazareth
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