  We consider the problem of finding an approximate second-order stationary
point of a constrained non-convex optimization problem. We first show that,
unlike the gradient descent method for unconstrained optimization, the vanilla
projected gradient descent algorithm may converge to a strict saddle point even
when there is only a single linear constraint. We then provide a hardness
result by showing that checking $(\epsilon_g,\epsilon_H)$-second order
stationarity is NP-hard even in the presence of linear constraints. Despite our
hardness result, we identify instances of the problem for which checking second
order stationarity can be done efficiently. For such instances, we propose a
dynamic second order Frank--Wolfe algorithm which converges to ($\epsilon_g,
\epsilon_H$)-second order stationary points in
${\mathcal{O}}(\max\{\epsilon_g^{-2}, \epsilon_H^{-3}\})$ iterations. The
proposed algorithm can be used in general constrained non-convex optimization
as long as the constrained quadratic sub-problem can be solved efficiently.
