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SIMULATION AND THE MONTE CARLO METHOD Episode 9

Tham khảo tài liệu 'simulation and the monte carlo method episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 220 SENSITIVITY ANALYSIS AND MONTE CARLO OPTIMIZATION g is the derivative of g that is the second derivative of f. The latter is given by g . V2 u E 5 X 4 3 X32 e-x3 u-1- - and can be estimated via its stochastic counterpart using the same sample as used to obtain Indeed the estimate of u is simplythe derivative of g at u. Thus an approximate 1 a confidence interval for u is u c g iu . This is illustrated in Figure 7.2 where the dashed line corresponds to the tangent line to g u at the point w 0 and 95 confidence intervals for g u and u are plotted vertically and horizontally respectively. The particular values for these confidence intervals were found to be -0.0075 0.0075 and 1.28 1.46 . Finally it is important to choose the parameter V under which the simulation is carried out greater than u . This is highlighted in Figure 7.3 where 10 replications of u are plotted for the cases V 0.5 and V 4. Figure 7.3 Ten replications of W u r are simulated under V 0.5 and V 4. In the first case the estimates of u v u u fluctuate widely whereas in the second case they remain stable. As a consequence u cannot be reliably estimated under V 0.5. Forv 4 no such problems occur. Note that this is in accordance with the general principle that the importance sampling distribution should have heavier tails than the target distribution. Specifically under V 4 the pdf of x3 has heavier tails than under V u whereas the opposite is true for V 0.5. In general let t and u denote the optimal objective value and the optimal solution of the sample average problem 7.48 respectively. By the law of large numbers Ể u v converges to u with probability 1 w.p.l as N 00. One can show 18 that under mild additional conditions and u converge w.p. 1 to their corresponding optimal objective value and to the optimal solution of the true problem 7.47 respectively. That is and u SIMULATION-BASED OPTIMIZATION OF DESS 221 are consistent estimators of their true counterparts t and u respectively. Moreover 18 .