Necessary optimality conditions and exact penalization for non-Lipschitz nonlinear programs (Q2413100)

The purpose of this paper is to study necessary optimality conditions and exact penalization for the following non-Lipschitz nonlinear program: \[ f(x) + \Phi(x) \longrightarrow \min,\text{ subject to }g(x) \leq 0,\, h(x) = 0, \] where \(f:\mathbb R^d \longrightarrow \mathbb R\), \(g : \mathbb R^d \longrightarrow \mathbb R^m\) are Lipschitz around the point of interest, and \(\Phi :\mathbb R^d\longrightarrow (-\infty, \infty]\) is an extended-valued lower semi-continuous function. The authors extend quasi-normality and the relaxed constant positive linear dependence condition and show that they are sufficient for Karush-Kuhn-Tucker conditions to be necessary for optimality. Moreover, exact penalization results for two special cases are derived. Including a non-Lipschitz term in the objective function significantly enlarges the applicability of standard nonlinear programs.