An optimization problem asks for decision variables that make an objective as small or as large as possible while respecting any constraints.

A minimization problem can be written as

$$\min_{x\in\mathcal F} f(x),$$

where $f$ is the objective function and $\mathcal F$ is the feasible set of allowed values.

A point in $\mathcal F$ is feasible. A point outside it is not a candidate solution, even if it gives a better objective value.

Constraints can be equalities such as

$$g(x)=0$$

or inequalities such as

$$h(x)\le0.$$

An unconstrained problem has $\mathcal F$ equal to the whole domain under consideration.

Optimization separates two questions: what choices are allowed, and how those choices are ranked. The same mathematical structure appears in design, estimation, control, scheduling and machine learning.