Consider a function Let be some subset of , which could be the entire . We denote by the standard Euclidean norm on .
A point is a local minimum of over if there exists an such that for all satisfying we have
If the inequality in (1.3) is strict for , then we have a strict local minimum. If (1.3) holds for all , then the minimum is global over . By default, when we say ``a minimum" we mean a local minimum. Obviously, a minimum need not be unique unless it is both strict and global.
The notions of a (local, strict, global) maximum are defined similarly. If a point is either a maximum or a minimum, it is called an extremum. Observe that maxima of are minima of , so there is no need to develop separate results for both. We focus on the minima, i.e., we view as a cost function to be minimized (rather than a profit to be maximized).