Changes

The ParSA library documentation gives the following equation which can be used to estimate performance of an MIR run

The ParSA library documentation gives the following equation which can be used to estimate performance of an MIR run

{|align=center

 

|<math> P(\chi_n \notin Cost_{min}) \sim \left(\frac{K}{n}\right)^{\alpha} </math>

{|width="80%"

|(1)

|align="right"|

<math> P(\chi_n \notin Cost_{min}) \sim \left(\frac{K}{n}\right)^{\alpha} </math>

|align="center" width="80"|(1)

|}

|}

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

{|align=center

{|width="80%"

|<math> \hat{p} = \frac{n_f + 1}{N + 2} </math>

|align="right"|

|(2)

<math> \hat{p} = \frac{n_f + 1}{N + 2} </math>

|align="center" width="80"|(2)

|}

|}

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

{|align=center

{|width="80%"

|<math> T=Rn </math>

|align="right"|

|(3)

<math> T=Rn </math>

|align="center" width="80"|(3)

|}

|}

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  

{|align=center

{|width="80%"

|<math> P_s = 1 - \left(\frac{K}{n}\right)^{R\alpha} </math>

|align="right"|

|(4)

<math> P_s = 1 - \left(\frac{K}{n}\right)^{R\alpha} </math>

|align="center" width="80"|(4)

|}

|}

{|width="80%"

{|width="80%"

|align="right"

|align="right"|

|<math> n = K e </math>

<math> n = K e </math>

|align="center" width="80"

|align="center" width="80"|(5)

|(5)

|}

|}

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>.