  Non-concave maximization has been the subject of much recent study in the
optimization and machine learning communities, specifically in deep learning.
Recent papers Ge et al, Lee et al (and references therein) indicate that first
order methods work well and avoid saddle points. Results as in Lee et al,
however, are limited to the \textit{unconstrained} case or for cases where the
critical points are in the interior of the feasibility set, which fail to
capture some of the most interesting applications. In this paper we focus on
\textit{constrained} non-concave maximization. We analyze a variant of a
well-established algorithm in machine learning called Multiplicative Weights
Update (MWU) for the maximization problem $\max_{\mathbf{x} \in D}
P(\mathbf{x})$, where $P$ is non-concave, twice continuously differentiable and
$D$ is a product of simplices. We show that MWU converges almost always for
small enough stepsizes to critical points that satisfy the second order KKT
conditions. We combine techniques from dynamical systems as well as taking
advantage of a recent connection between Baum Eagon inequality and MWU
(Palaiopanos et al).
