Nonregular mixed-constrained optimization problems

We present higher-order necessary optimality conditions for mixed-constrained problems. We study the case when the constraints of the problems are not assumed to be regular at a solution. We introduce some new generalized regularity conditions and we obtain Karush–Kuhn–Tucker type necessary optimality conditions. Also, we discuss the connections between these new regularity conditions and the Lagrange multipliers set. Some examples are presented to illustrate our results.