Global Minima

A global minimum of $f$ on a set $D$ is a point $\mathbf{a}^* \in D$ such that $f(\mathbf{a}^*) \leq f(\mathbf{x})$ for all $\mathbf{x} \in D$. Finding the global minimum is generally much harder than finding local minima or critical points.

Every global minimum is a local minimum (and hence a critical point for differentiable $f$ on an open domain), but not vice versa. For convex functions on convex domains, every local minimum is global — this is why convex optimization problems are tractable.

For non-convex functions, the only guaranteed strategy is to find all critical points and compare their values, or to use global optimization algorithms (branch and bound, simulated annealing, evolutionary methods). In practice, many problems are solved by assuming the objective is approximately convex near the solution.