Revisiting approximate minimality and approximate criticality in scalar optimization with constraints

This paper examines various questions related to approximate minimality and approximate 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 relationship between approximate criticality and genuine minimality of perturbed functions. A key tool in deriving these results is the use of well-known penalization functions.