local optimum

A point that beats every nearby feasible point, but not necessarily every feasible point overall.

A feasible $x^*$ is a local minimum if $f(x^*)\le f(x)$ for every feasible $x$ within some neighborhood of $x^*$ (a small enough ball around it) — it need not be the best point in the entire feasible region, only the best point nearby. Local maximum is defined the same way with the inequality reversed. "Local optimum" covers either case generically.