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By I. D. L. Bogle, J. Zilinskas

ISBN-10: 981256909X

ISBN-13: 9789812569097

ISBN-10: 9812772952

ISBN-13: 9789812772954

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Additional resources for Computer Aided Methods in Optimal Design and Operations (Series on Computers and Operations Research)

Sample text

In agglomeration step, if necessary, tasks are combined into larger tasks to improve performance and to reduce development and communication costs. This step is not necessary for parallel BB algorithms implemented in the template. Then each subtask is mapped or assigned to a processor. During this step we try to minimize the computation time by preserving a good load balance among processors. If the BB template is used for sequential programming, then in order to get a parallel variant of the program, the user has to select one of proposed parallel algorithms and compile parallel version of the program.

The General Quadratic Case Consider the problem of finding the minimum value x* of a quadratic function f(x) = -x1Ax - xTb where A is a positive definite symmetric matrix with eigenvalues 0 < m = Ai < A2 < ... < \d = M < 00. Without loss of generality assume that /Ai 0 . . 0 \ 0 A2 0 A = . . , (3) \ 0 0 ... xj since a change of variables can always achieve this form. The gradient of f(x) is g{x) = Ax — b. Rate of Convergence of the Steepest Descent Optimisation Algorithm with Relaxation 51 We can also assume that x* = 0 (otherwise the substitution x — x* —» x can be made).

Zhigljavsky, Branch and probability bound methods for global optimization. Informatica 1(1), 125-140 (1990). 12. H. Chen, Estimation of the location of the maximum of a regression function using extreme order statistics, Journal of multivariate analysis 57, 191-214 (1996). 13. P. Hall, On estimating the endpoint of a distribution, Annals of Statistics 10, 556-568 (1982). 14. S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (2), 327-343 (1993).

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