Formal Formulation of Optimization
If we want to write down an optimization problem…
\begin{align} \min_{x} f(x) \\ s.t: x \in \mathcal{X} \end{align}
where:
- \(x\): “design point” (usually a vector representing the thing you are trying to optimize)
- which is an assignment of “design variable” to values (width, height, etc.)
- \(\mathcal{X}\): a feasible set of design points that satisfy the constraint
- \(f: \mathcal{X} \to \mathbb{R}\): the objective function, which
- \(x^{*}\): the minimizer—one (or many) points that satisfy the minimization scheme
Conventionally, we minimize.
constraint
Frequently, the feasible set consists of some linear expressions which forces the correct constraints; for instance:
\begin{align} \min_{x_1, x_2} &\ f(x_1, x_2) \\ s.t. &\begin{cases} x_1 \geq 0 \\ x_2 \geq 0 \\ x_1 + x_2 \leq 1 \end{cases} \end{align}
You can then shade out the areas of the entire problem space which doesn’t satisfy the consraints, individually.