We will now turn to consider mathematical optimization as a computational problem.

Consider the usual:

$$ \begin{aligned} \min ~& f( x) \\ \text{s.t.} ~~& x \in S \subseteq {\R}^d \end{aligned} $$

• assume that the minimum is attained at some $x^\in S*$ and denote $f^ = f(x^)$
• as usual, we also assume $\text{dom}(f) \subseteq \R^d$ is open and $S \subseteq \text{dom}(f)$ is bounded and closed

Can it be solved algorithmically?

• What does it mean to “solve” the problem?
• How to specify the objective $f$ to the algorithm?
• How to specify the domain $S$ to the algorithm?
• What assumptions to place on $f$ and $S$ for the problem to make sense?
• When can we say that a problem is efficiently solvable? That an algorithm is efficient?

Many of these are straightforward for discrete problems, but here our focus in on continuous problems.

Solution concepts

Exact solution