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Optimization Methods - Notes on Use with Boomer

The damping Gauss-Newton method provides progress output.

 Loop =     1
 Damp =     1
 P ( 1) =       .5808    
 P ( 2) =       .1106    
 P ( 3) =       2.101    
 P ( 4) =       39.64    
 WSS =        55.5363    


 Loop =     2
 Damp =     1
 P ( 1) =       .4375    
 P ( 2) =       .1319    
 P ( 3) =       2.199    
 P ( 4) =       42.54    
 WSS =        10.6959    

Each iteration is represented as a loop number. Damp numbers higher than 2 or 3 indicate that the surface may be irregular.

The Nelder-Mead (Simplex) method provides progress output.

 Loop   1 - 
   1>   717.5      2>   643.2      3>   386.5      4>   973.2      5>   797.9    
 Loop   2 - 
   1>   717.5      2>   643.2      3>   386.5      4>   127.4      5>   797.9    
 Loop   3 - 
   1>   717.5      2>   643.2      3>   386.5      4>   127.4      5>   120.6    
 Loop   4 - 
   1>   58.02      2>   643.2      3>   386.5      4>   127.4      5>   120.6    
 Loop   5 - 
   1>   58.02      2>   38.82      3>   386.5      4>   127.4      5>   120.6   

From line to line there should be one value, the highest, WSS reduced from the previous line.

If selecting the Simplex or Simplex->Damping GN method a convergence criteria (PC) must be entered. If the Gauss-Newton, Damping Gauss-Newton, or Marquardt method are selected a DT value must entered as well. When editing .BAT file there is one more line with these methods. The DT value specifies the step-size used to create the partial differentials used in the optimization process. In both cases (PC and DT) the default values usually work well.

 FITTING METHODS

 0) Gauss-Newton
 1) Damping Gauss-Newton
 2) Marquardt
 3) Simplex
 4) Simplex->Damping GN

 Enter Choice (0-4)      4

 Enter PC for convergence (0.00001)      .000

or

 FITTING METHODS

 0) Gauss-Newton
 1) Damping Gauss-Newton
 2) Marquardt
 3) Simplex
 4) Simplex->Damping GN

 Enter Choice (0-4)      1

 Enter DT for Jacobian (0.001)      .000    

 Enter PC for convergence (0.00001)      .000    

The first three methods are numerically intensive, relative to the simplex method, thus there may be occasion when poorly defined calculations may produce 'fatal' errors, for example divide by zero or other math error. Better initial estimates or use of the less numerically intensive simplex method may be useful.

Repeated Runs using the Simplex Method

Another advantage of the simplex method as implemented in Boomer is that once the initial estimates (first point on the simplex) are specified the other points on the simplex are randomly generated. This means that each time a problem is run a different initial simplex is used and effectively different initial estimates are used with each run. This can be set-up automatically by changing the last line of the .BAT file. Consider,

    0                             AUC line number                    
    2                             Continue,save,plot,both            
   -2                             wls,bayes,sim,irwls,sim+error,grid 

If the -2 on the last line is replaced with a -4 and followed, on a new line, with the name of the xxxx.BAT file without the .BAT, Boomer will rerun the .BAT file repeatedly using a different initial simplex each time until the user stops the program with a control-C (or similar).

    0                             AUC line number                    
    2                             Continue,save,plot,both            
   -4                             wls,bayes,sim,irwls,sim+error,grid
xxxx
The .OUT file will now contain the results of multiple runs and the WSS (and other results) from each run can be examined. Typically there will be a majority of runs that converge to the same 'global' minimum with a few that might stops with higher WSS values.

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