Converting goals to constraints

The way goals are converted into constraints depends on the type of goal. Details on the type of goals can be found in Goals.

Minimization goals

Suppose we have a minimization goal of the form

\[\text{minimize } f(x)\]

for the current priority, and we find a solution \(x^*\). For the next priority, this goal is converted into a constraint of the form

\[\text{subject to } f(x) \leq f(x^*).\]

Target goals

Suppose we have a target goal of the form

\[\begin{split}\text{minimize } & \epsilon^r, \\ \text{subject to } & g_{low}(\epsilon) \leq g(x) \leq g_{up}(\epsilon), \\ \text{and } & 0 \leq \epsilon \leq 1,\end{split}\]

for the current priority, and we find a solution \(x^*\) and \(\epsilon^*\). For the next priority, this target goal is converted into either a hard constraint or a soft constraint.

Converting to a hard constraint (default)

When converting a target goal into a hard constraint, the target goal is converted into the followingng constraint.

\[\text{subject to } g_{low}(\epsilon^*) \leq g(x) \leq g_{up}(\epsilon^*).\]

Converting to a soft constraint

When converting a target goal into a soft constraint, the target goal is replaced by the following constraints.

\[\begin{split}\text{subject to } & \epsilon^r \leq (\epsilon^*)^r, \\ \text{and } & g_{low}(\epsilon) \leq g(x) \leq g_{up}(\epsilon), \\ \text{and } & 0 \leq \epsilon \leq 1.\end{split}\]

In other words, we keep the epsilon variables and original constraints, but only replace the epsilon minimisation goal by the constraint \(\epsilon^r \leq (\epsilon^*)^r\).