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In continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether   (a)   a given feasible solution is not a local minimum, and(b)   the objective function is not bounded below on the set of feasible solutions.We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.

Some NP-complete problems in quadratic and nonlinear programming