In this paper, we study KKT optimality conditions for constrained nonlinear programming problems and strong and Mordukhovich stationarities for mathematical programs with complementarity constraints using $l_p$ penalty functions, with $0\leq p\leq1$. We introduce some optimality indication sets by using contingent derivatives of penalty function terms. Some characterizations of optimality indication sets are obtained by virtue of the original problem data. We show that the KKT optimality condition holds at a feasible point if this point is a local minimizer of some $l_p$ penalty function with $p$ belonging to the optimality indication set. Our result on constrained nonlinear programming includes some existing results from the literature as special cases.