  Hidden convex optimization is such a class of nonconvex optimization problems
that can be globally solved in polynomial time via equivalent convex
programming reformulations. In this paper, we focus on checking local
optimality in hidden convex optimization. We first introduce a class of hidden
convex optimization problems by jointing the classical nonconvex trust-region
subproblem (TRS) with convex optimization (CO), and then present a
comprehensive study on local optimality conditions. In order to guarantee the
existence of a necessary and sufficient condition for local optimality, we need
more restrictive assumptions. To our surprise, while (TRS) has at most one
local non-global minimizer and (CO) has no local non-global minimizer, their
joint problem could have more than one local non-global minimizer.
