Lagrange multipliers and calmness conditions of order p

In this paper, by assuming that a non–Lipschitz penalty function is exact, new conditions for the existence of Lagrange multipliers are established for an inequality and equality–constrained continuously differentiable optimization problem. This is done by virtue of a first–order necessary optimality condition of the penalty problem, which is obtained by estimating Dini upper–directional derivatives of the penalty function in terms of Taylor expansions, and a Farkas lemma. Relations among the obtained results and some well–known constraint qualifications are discussed.