A convex optimization problem minimizes a convex objective over a convex feasible set.

Its defining advantage is that every local minimum is global. Suppose $x_$ were a local minimum but another feasible point $y$ had $f(y)<f(x_)$. Convexity would make every point sufficiently close to $x_*$ on the segment toward $y$ feasible and give it a lower objective value, contradicting local optimality.

A convex problem can have several global minimizers. If the objective is strictly convex on a convex feasible set, however, it can have at most one minimizer.

Convexity does not automatically make a problem computationally cheap, but it removes deceptive local minima and gives optimization algorithms unusually strong global guarantees.

Linear programs, linear least-squares problems and many regularized objectives are important examples of convex optimization.