Optimization Problem Formulation

We define the standard form of the optimization problem as:

Minimize $f_{i, x} (x) \forall i = 1, 2, ... m$ such that $g_{k} (x) < G_{k} \forall k = 1, 2, ... n$ .

Given weights $w_{i} \forall i = 1, 2, ... p$ for the $p$ parameters; these weights are computed from the priorities specified for the parameters and the weights associated with the groups of parameters.

Modified Unconstrained Single Objective Formulation

We aim to minimize $s (x)$ , where $s (x) = o b j e c t i v e (x) + p e n a l t y (x)$ .

Objective Function

The objective function is as follows:

o b j e c t i v e (x) = \frac{1}{m} \sum_{i = 1}^{m} w_{i} \frac{f_{i} (x) - f_{i, \min} |_{D A T A S E T}}{f_{i, \max} |_{D A T A S E T} - f_{i, \min} |_{D A T A S E T}},

where $f_{i, \min} |_{D A T A S E T}$ and $f_{i, \max} |_{D A T A S E T}$ are the minimum and maximum values of the objective function seen in the original dataset. Target objectives are treated as the square of the deviation from the target.

Penalty Function

Results Analytics uses a step and scaled violation as the penalty function. The step is used to introduce a clear separation between feasible (but not necessarily optimal) and optimal (but slightly infeasible) designs. The penalty function is as follows:

p e n a l t y (x) = {\begin{matrix} 0 \\ P E N A L T Y_S T E P + P E N A L T Y_M U L T I P L I E R * t s v (g_{i}, x) \end{matrix} \begin{matrix} i f g_{i} (x) < G_{i} \forall i \\ o t h e r w i s e \end{matrix}

where

t s v (g_{i}, x) = \sum_{i = 1}^{n} {w'}_{i} * s v_{i} (x),

{w'}_{i} = {\begin{matrix} M a x (P E N A L T Y_F O R_M H, w_{i}) \\ w_{i} \end{matrix} \begin{matrix} i f p_{i} i s M H (m u s t h a v e p r i o r i t y) \\ o t h e r w i s e \end{matrix},

and

s v_{i} (x) = {(\frac{v_{i} (x)}{v_{i, \max} |_{D A T A S E T}})}^{P E N A L T Y_E X P O N E N T} .

$v_{i, \max} |_{D A T A S E T}$ is the maximum violation for the constraint $i$ in the original data set, and $v_{i} (x) = | g_{i} (x) - G_{i} |$ for the inequality constraints.

Default Values for Constants

The constants above have the following default values:

PENALTY_STEP = m, the number of objectives
PENALTY_MULTIPLIER = 1000
PENALTY_FOR_MH = 1000
PENALTY_EXPONENT = 2