A note on stationarity in constrained optimization

Abstract: Minimizing a smooth function f on a closed subset C leads to different notions of stationarity: Fr{\'e}chet stationarity, which carries a strong variational meaning, and criticality, which is defined through a closure process and involves the notion of limiting, or Mordukovitch, subdifferential. The latter is an optimality condition which may loose the variational meaning of Fr{\'e}chet stationarity in some settings. The purpose of this note is to illustrate that, while criticality is the appropriate notion in full generality, Fr{\'e}chet stationarity is typical in practical scenarios.We gather two results to illustrate this phenomenon. These results are essentially known and, our goal is to provide consize self contained arguments in the constrained optimization setting. 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 Fr{\'e}chet stationary. Second we prove that for small step-sizes, all the accumulation points of the projected gradient algorithm are Fr{\'e}chet stationary, with an explicit global quadratic estimate of the remainder, avoiding potential critical points that are not Fr{\'e}chet stationary, and some bad local minima.