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<math> P(\chi_n \notin Cost_{min}) \sim \left(\frac{K}{n}\right)^{\alpha} </math>
<math> P(\chi_n \notin Cost_{min}) \sim \left(\frac{K}{n}\right)^{\alpha} </math>
where ''P'' is the probability of non-convergence, <math>\chi_n</math> is a solution of a run of length ''n'', <math>Cost_{min}</math> is the minimum acceptable solution, and ''K'' and <math>\alpha</math> are problem specific parameters. ''K'' and <math>\alpha</math> can be determined by plotting the Bayesian estimator for ''P'' versus ''n'' on a log scale and determining the slope and y-intercept. The expression for the Bayesian estimator <math> \hat{p} </math> is given by
where ''P'' is the probability of non-convergence, <math>\chi_n</math> is a solution of a run of length ''n'', <math>Cost_{min}</math> is the minimum acceptable solution, and ''K'' and <math>\alpha</math> are problem specific parameters. ''K'' and <math>\alpha</math> can be determined by plotting the Bayesian estimator for ''P'' versus ''n'' on a log scale and determining the slope and y-intercept. The expression for the Bayesian estimator <math> \hat{p} </math> is given by
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<math> \hat{p} = \frac{n_f + 1}{N + 2} </math>
<math> \hat{p} = \frac{n_f + 1}{N + 2} </math>
The optimal run length can be determined now that the ''K'' and <math>\alpha</math> parameters are known. Given that the the total number of steps ''T'' is given by the following expression
The optimal run length can be determined now that the ''K'' and <math>\alpha</math> parameters are known. Given that the the total number of steps ''T'' is given by the following expression
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|(3)
<math> T=Rn </math>
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where ''R'' is the number of runs and ''n'' is the run length. The probability of obtaining one successful run out of ''R'' runs is given by
where ''R'' is the number of runs and ''n'' is the run length. The probability of obtaining one successful run out of ''R'' runs is given by
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|(4)
<math> P_s = 1 - \left(\frac{K}{n}\right)^{R\alpha} </math>
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Using the method of Lagrange multipliers, the optimal step number can be found to be
Using the method of Lagrange multipliers, the optimal step number can be found to be
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<math> n = K e </math>
<math> n = K e </math>
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and given the ''K'' parameter determined from the ''alpha'' = 0.5 run, the optimal step number can be determined to be <math> (1.44 \pm 0.65) \times 10^5 </math>.
and given the ''K'' parameter determined from the ''alpha'' = 0.5 run, the optimal step number can be determined to be <math> (1.44 \pm 0.65) \times 10^5 </math>.