  The local nonglobal minimizer of trust-region subproblem, if it exists, is
shown to have the second smallest objective function value among all KKT
points. This new property is extended to $p$-regularized subproblem. As a
corollary, we show for the first time that finding the local nonglobal
minimizer of Nesterov-Polyak subproblem corresponds to a generalized eigenvalue
problem.
