Revisiting approximate minimality and approximate criticality in scalar optimization with constraints

This paper examines various questions related to approximate minimality and approx- imate criticality in constrained scalar optimization. We establish necessary optimality conditions for the approximate minimality of a lower semicontinuous (and therefore non-Lipschitz) function under a set constraint. Additionally, we explore the relation- ship between approximate criticality and genuine minimality of perturbed functions. A key tool in deriving these results is the use of well-known penalization functions.