2ND-ORDER CONDITIONS FOR EXTREMUM PROBLEMS WITH NONREGULAR EQUALITY CONSTRAINTS

Combining results of Avakov about tangent directions to equality constraints given by smooth operators with results of Ben-Tal and Zowe, we formulate a second-order theory for optimality in the sense of Dubovitskii-Milyutin which gives nontrivial conditions also in the case of equality constraints given by nonregular operators. Second-order feasible and tangent directions are defined to construct conical approximations to inequality and equality constraints which within a single construction lead to first- and second-order conditions of optimality for the problem also in the nonregular case. The definitions of second-order feasible and tangent directions given in this paper allow for reparametrizations of the approximating curves and give approximating sets which form cones. The main results of the paper are a theorem which states second-order necessary condition of optimality and several corollaries which treat special cases. In particular, the paper generalizes the Avakov result in the smooth case.