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

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Sen, Statistical approximations for stochastic linear programming problems, Annuals of Operations Research 85, 173-192 (1999). 7. V. S. Mikhalevitch, A. M. Gupal and V. I. Norkin, Methods of Nonconvex Optimization, Nauka, Moscow (in Russian) (1987). 8. H. J. Kushner and G. G. , Heidelberg, Berlin (2003). 9. K. Marti, Descent stochastic quasigradient methods, In: Yu. Ermolyev and R. ), Numerical Techniques for Stochastic Optimization, Springer-Verlag, Berlin, pp. 393-400 (1988). 10. K. Marti, Optimal semi-stochastic approximation procedures, ii.

Processors can share UB{D). When new value of the upper bound is found, it is broadcasted to the other processors. In order not to stop calculations, this exchange is performed asynchronously. These modifications of the BB algorithm will be called SJP SE and RJP SE, depending on the rule to distribute the initial job pool. fr). Figures 2 and 3 present the calculation times. Implementation of Parallel Optimization Algorithms / X ••. // - L SJP - - - SJPSE -- RJP RJPSE ^-^—^s. \ 32 64 128 256 _ o • 1 2 4 8 27 16 32 64 128 256 Processors Figure 2.

One way of proving that an expression is non-negative is to relate it to a variance of a random variable, as it is known that variances are always non-negative. Consider the variance V = var(a£ + b£2) Rate of Convergence of the Steepest Descent Optimisation Algorithm with Relaxation 55 where f is the random variable with distribution P and a, b are some parameters we shall choose. V = var(at + b£2) = E(a£ + fo£2)2 - [E(a£ + 6£2)]2 = a2E(e) + 2abE{e) + b2E(^) - ([aE® + &£(£2)])2 = a2fi2 + 2abnz + b2/u4 - (a/zi + &M2)2 • Subtract the variance V from U and consider this as a function of a, b: F(a, b) = U -V = 47VM3M2-7W-4MI + 4 / i i 6 7 4 M i W 7 + 4/ii4/X2 6 -47/ii 3 /x 3 + 72/i4/Ui2 - /Ui2M2272 + 4/x 1 4 // 2 7 + 47 2 /ii 3 /x 3 +7V4M2 - 2 7 3 /i 4/ ui 2 + 5/u 1 2 // 2 2 7 3 -8/X1V27 2 - 2 73//iM3M2 - 7 V 2 3 - a2M2 - 2 a&^3 - 62/i4 + a 2 /ii 2 + 2 a/iib^ + b2fi22 • Let us attempt to select a, b so that F(a, 6) = 0, which would mean that U = V .

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