By Luc Pronzato, Anatoly Zhigljavsky
This edited quantity, devoted to Henry P. Wynn, displays his huge variety of study pursuits, focusing specifically at the purposes of optimum layout idea in optimization and data. It covers algorithms for developing optimum experimental designs, basic gradient-type algorithms for convex optimization, majorization and stochastic ordering, algebraic facts, Bayesian networks and nonlinear regression. Written by way of prime experts within the box, each one bankruptcy incorporates a survey of the present literature besides large new fabric. This paintings will entice either the professional and the non-expert within the parts lined. by way of attracting the eye of specialists in optimization to big interconnected components, it may support stimulate extra learn with a possible impression on functions.
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Extra resources for Optimal Design and Related Areas in Optimization and Statistics (Springer Optimization and Its Applications)
1959). On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Annals of the Institute of Statistical Mathematics, Tokyo, 11, 1–16. Barzilai, J. and Borwein, J. (1988). Two-point step size gradient methods. IMA Journal of Numerical Analysis, 8, 141–148. Fedorov, V. (1972). Theory of Optimal Experiments. Academic Press, New York. Forsythe, G. (1968). On the asymptotic directions of the s-dimensional optimum gradient method. Numerische Mathematik, 11, 57–76.
Ks−1 ) [Mks,1 ]−1 43 ⎞ t ⎜ .. ⎟ ⎝ . ⎠ ts and direct calculations give 1 μk1 . . μks−1 μk1 μk2 . . μks .. . . . .. μks μks+1 . . μk2s−1 Qks (t) = |Mks,1 | 1 t .. 6) where, for any square matrix M, |M| denotes its determinant. The derivation of the updating rule for the normalized gradient zk relies on the computation of the inner product (gk+1 , gk+1 ). From the orthogonality property of gk+1 to gk , Agk , . . , As−1 gk we get (gk+1 , gk+1 ) = (gk+1 , γsk As gk ) = γsk (Qks (A)As gk , gk ) 1 μk1 .
Theorem 1 below shows that if the relaxation coeﬃcient ε is either small (ε < 4M m/(M + m)2 ) or large (ε > 1), then for almost all starting points the algorithm asymptotically behaves as if it has started at the worst possible initial point. However, for some values of ε the rate does not attract to a constant value and often exhibits chaotic behaviour. 5) is shown in Fig. 1 where we display the asymptotic rates in the case M/m = 10. In this ﬁgure and all other ﬁgures in this chapter we assume that d = 100 and all the eigenvalues are equally spaced.