Generic Fréchet stationarity in constrained optimization

Minimizing a smooth function f on a closed subset C leads to different notions of stationarity: Fréchet stationarity, which carries a strong variational meaning, and criticallity, which is defined through a closure process. The latter is an optimality condition which may loose the variational meaning of Fréchet stationarity in some settings. We show that, while criticality is the appropriate notion in full generality, Fréchet stationarity is typical in practical scenarios. This is illustrated with two main results, first we show that if C is semi-algebraic, then for a generic smooth semi-algebraic function f , all critical points of f on C are actually Fréchet stationary. Second we prove that for small step-sizes, all the accumulation points of the projected gradient algorithm are Fréchet stationary, with an explicit global quadratic estimate of the remainder, avoiding potential critical points which are not Fréchet stationary, and some bad local minima.