By B. Torsney (auth.), Luc Pronzato, Anatoly Zhigljavsky (eds.)
ISBN-10: 0387799354
ISBN-13: 9780387799353
ISBN-10: 0387799362
ISBN-13: 9780387799360
This edited quantity, devoted to Henry P. Wynn, displays his extensive variety of analysis pursuits, focusing particularly at the purposes of optimum layout concept in optimization and data. It covers algorithms for developing optimum experimental designs, normal gradient-type algorithms for convex optimization, majorization and stochastic ordering, algebraic facts, Bayesian networks and nonlinear regression. Written through major experts within the box, each one bankruptcy includes a survey of the present literature in addition to great new fabric.
This paintings will attract either the expert and the non-expert within the components lined. by way of attracting the eye of specialists in optimization to big interconnected components, it's going to aid stimulate additional learn with a possible impression on applications.
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Extra info for Optimal Design and Related Areas in Optimization and Statistics
Example text
K2s−2 γsk = . 2 The Optimum s-Gradient Algorithm as a Sequence of Transformations of a Probability Measure When H = Rd , we can assume that A is already diagonalized, with eigenvalues 0 < m = λ1 ≤ λ2 ≤ · · · ≤ λd = M , and consider [zk ]2i , with [zk ]i the d ith component of zk , as a mass on the eigenvalue λi (note that i=1 [zk ]2i = μk0 = 1). Define the discrete probability measure νk supported on (λ1 , . . 3). Remark 3. When two eigenvalues λi and λj of A are equal, their masses [zk ]2i and [zk ]2j can be added since the updating rule is the same for the two components [zk ]i and [zk ]j .
2000), p. 175) one can see that the masses ξ (k) (m) are bounded away from 0; that is, ξ (k) (m) ≥ c for some c > 0 and all k, as long as ξ0 (m) > 0. Therefore, the point m always belongs to the support of the measure ξ∗ . If 0 < ε < 2m/(m + M ) then one can easily check that ξ∗ must be concentrated at a single point (otherwise U = U (ξ∗ ) = 0); this point is necessarily m and therefore the limiting design ξ∗ coincides with the optimal design ξ ∗ . 25). One of the implications of the theorem is that if the relaxation parameter is either too small (ε < 2m/(m + M )) or too large (ε > 2M/(m + M )) then the rate of the steepest-descent algorithm with relaxation becomes worse than Rmax , the worst-case rate of the standard steepest-descent algorithm.
Ks−1 )(Mks−1,0 )−1 ⎜ . ⎟ = (Mks,0 )−1 s+1,s+1 ⎝ .. d. errors (or to the determination of a c-optimal measure on [m, M ] with c = (0, . . , 0, 1) ). This measure is uniquely defined, see (Kiefer and Wolfowitz, 1959, p. 16) and the weights are ξ ∗ (−1) = ξ ∗ (1) = 1/(2s) , ξ ∗ (cos[jπ/s]) = 1/s , 1 ≤ j ≤ s − 1 . The transformation t ∈ [−1, 1] → z = (M + m)/2 + t(M − m)/2 ∈ [m, M ] gives the measure νs∗ on [m, M ]. 24), see Kiefer and Wolfowitz (1959, p. 283). The monotonicity of the sequence (qk ), together with Theorem 3, implies the following analogue to Corollary 3.
Optimal Design and Related Areas in Optimization and Statistics by B. Torsney (auth.), Luc Pronzato, Anatoly Zhigljavsky (eds.)
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