(Mathematical, continuous) Optimization?

Unconstrained optimization problem:

$$ \operatorname*{minimize}_{x \in \R^d} \; f(x) $$

• Candidate solutions/parameters/variables $x \in \R^d$
• Objective function $f : \R^d \to \R$

Constrained optimization problem:

$$ \begin{aligned} \operatorname*{minimize}_{x \in \R^d} \;\; & f(x) \\ \text{subject to } \;\;& c_i(x) \leq 0, \; i = 1, \ldots, m \\ & h_j(x) = 0, \; j = 1, \ldots, n \end{aligned} $$

• Inequality constraints: $c_i(x) \leq 0$, $i = 1, \ldots, m$
• Equality constraints: $h_j(x) = 0$, $j = 1, \ldots, n$

Can be written more abstractly as:

$$ \operatorname*{minimize}_{x \in S} \; f(x) $$

• $S \subseteq R^d$ is the optimization domain, e.g.:

  $$ ⁍

  $$

Why Optimization?

Optimization is used as a primary computational tool in many fields: