By Pronzato L., Zhigljavsky A. (eds.)
This edited quantity, devoted to Henry P. Wynn, displays his vast variety of study pursuits, focusing particularly at the purposes of optimum layout idea in optimization and records. It covers algorithms for developing optimum experimental designs, normal gradient-type algorithms for convex optimization, majorization and stochastic ordering, algebraic records, Bayesian networks and nonlinear regression. Written by way of prime experts within the box, each one bankruptcy encompasses a survey of the present literature in addition to immense new fabric. This paintings will entice either the professional and the non-expert within the parts lined. via attracting the eye of specialists in optimization to special interconnected components, it may support stimulate extra study with a possible impression on functions.
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Extra resources for Optimal design and related areas in optimization and statistics
Example text
4 for the steepest-descent algorithm with relaxation. 34 R. Haycroft et al. 6 Fig. 7. 5 Fig. 8. 9 is similar to Fig. 01]. In Fig. 99, see Sect. 7, and (d) the algorithm based on A-optimality criterion, see Sect. 9. 29); that is, as the ratios Rmin /R, where R is the asymptotic rate of the respective algorithm. 01 Fig. 9. 01] Fig. 10. Efficiency relative to Rmin for various algorithms, varying In Fig. 11, we compare the asymptotic rates (in the form of efficiencies with respect to Rmin ) of the following gradient algorithms: (a) α-root algorithm with optimal value of α; (b) steepest descent with optimal value of the relaxation coefficient ε; (c) Cauchy–Barzilai–Borwein method (CBB) as defined in (Raydan and Svaiter, 2002); (d) Barzilai–Borwein method (BB) as defined in (Barzilai and Borwein, 1988).
The conclusion is that this algorithm has an extremely fast rate when α is slightly larger than 1. The α-root algorithm with relaxation (this class of algorithms includes the steepest-descent and square-root algorithms with relaxation) is an obvious generalization of the algorithm of steepest descent with relaxation. Its asymptotic behaviour is also similar: for a fixed α, for very small and very large values of the relaxation parameter ε, the algorithm either diverges or converges with the rate ≥ Rmax .
Asymptotic rate of convergence as a function of ε for the steepest-descent algorithm with relaxation ε Theorem 1. Assume that ε is such that either 0 < ε < 4M m/(m + M )2 or ε > 1. Let ξ0 be any non-degenerate probability measure with support {λ1 , . . 24) where pi are the masses ξ (k) (λi ). Then the following statements hold: • For any starting point x0 , the sequence Φk = Φ(M (ξ (k) )) monotonously increases (Φ0 ≤ Φ1 ≤ · · · ≤ Φk ≤ · · · ) and converges to a limit limk→∞ Φk . 2. Proof. Note that Φk ≤ 0 for all ξ (k) if 0 < ε < min ξ 4M m μ21 (ξ) = 2 .