Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.

A convex optimization problem is defined by two ingredients:

The goal of the problem is to find some ${\displaystyle \mathbf {x^{\ast }} \in C}$ attaining

In general, there are three options regarding the existence of a solution:

A convex optimization problem is in standard form if it is written as

where:

The feasible set ${\displaystyle C}$ of the optimization problem consists of all points ${\displaystyle \mathbf {x} \in {\mathcal {D}}}$ satisfying the inequality and the equality constraints. This set is convex because ${\displaystyle {\mathcal {D}}}$ is convex, the sublevel sets of convex functions are convex, affine sets are convex, and the intersection of convex sets is convex.

Many optimization problems can be equivalently formulated in this standard form. For example, the problem of maximizing a concave function ${\displaystyle f}$ can be re-formulated equivalently as the problem of minimizing the convex function ${\displaystyle -f}$. The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem.