By I. D. L. Bogle, J. Zilinskas
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Additional info for Computer Aided Methods in Optimal Design and Operations (Series on Computers and Operations Research)
Sample text
Initial search space is subsequently divided into smaller subspaces. Then each subspace is evaluated trying to find out if it can contain optimal solution. For this purpose a lower bound for objective function is calculated over the subspace and compared with upper bound for the minimum value. If the lower bound for objective function over the subspace is larger than the upper bound for minimum value, the subspace cannot contain the global minimizer and therefore it is rejected from further search.
First let us check that /U0 = YlPi = 1 • Indeed, 1 ^ 2c*yUi + a 2 /X2 ~ ^ 1 - 2a/ii + a 2 /x 2 (1 - aAi)2Pi 1 - 2a/xi 4- a ^2 1. /i x and /x2 expressed through the moments of P are then d 2a/i! + a 2 /i2 1 A*2 = iS= lA * p * = r - 2a/^i + a 2 /i2 jUi - 2a/U2 4- a fis ju2 - 2a^ 3 + a /u4 A rate of convergence at iteration k in this family of algorithms is defined by Ak) (g(fc+p>g(fc+i)) (gW,gW) _ (gW, g (fc))-2a(AffW,ffW)+a 2 (^V f c ),ffW) — 1- 2a/ii + a /12 • (gW,gW) (10) This rate corresponds to the denominator of the updating formula (7).
Stochastic procedures for solving problems of this kind are often considered and two ways are used to achieve the convergence of developed methods. The first one leads to the class of methods of stochastic approximation. 9'10 However, the methods of stochastic approximation converge rather slowly and, besides, it is not so clear how to terminate the process of stochastic approximation. The second way to ensure the convergence in stochastic optimisation is related to application of the methods of a relative stochastic gradient error.