About Objective Functions in the Approximation Loop and Pointer-2 Strategies

The value of the objective function is calculated as a sum of all objective components with corresponding weight and scale factors:

$(O b j e c t i v e = S u m ({O B J}_{i} \times W_{i} / S_{i}),$

where $O B J i$ is the contribution to the objective function of the $i - t h$ objective component (parameter), $W i$ is the corresponding weight factor, and $S i$ is the corresponding scale factor.

If the direction of a specific objective parameter is “minimize,” its contribution to the objective function equals the parameter value itself:

${O B J}_{i} = {P a r m}_{i}$

If the direction of a specific objective parameter is “maximize,” its contribution to the objective function equals the parameter value multiplied by –1 to reverse the direction, because the Optimization component always minimizes the objective function:

${O B J}_{i} = {P a r m}_{i} = - 1.0 \times {P a r m}_{i}$

If the direction of a specific objective parameter is “target,” its contribution to the objective function is calculated as follows:

${O B J}_{i} = {S q r t [(P a r m}_{i} {- T a r g e t}_{i})^{2} + 0.04] - 0.2$

The second component of the ObjectiveAndPenalty parameter is the penalty function, calculated as follows:

$P e n a l t y = P e n a l t y B a s e + P e n a l t y M u l t i p l i e r \times S u m {(V i o l a t i o n}_{i} \times W_{i} / S_{i}) \land P e n a l t y E x p o n e n t$

where $V i o l a t i o n i$ is the $i - t h$ constraint violation value, $W i$ is the corresponding weight factor, and $S i$ is the corresponding scale factor. The PenaltyBase option is set to zero if no constraints are violated. The default value is 0.0.