On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints

For nonconvex optimization problems in R(n), a counterexample was recently given by Anitescu [SIAM J. Optim., 10 (2000), pp. 1116-1135] that showed that the Mangasarian-Fromovitz constraint qualification (MFCQ) is not sufficient for the classical necessary second-order optimality conditions to hold. We prove these optimality conditions with the assumptions that the set of Lagrange multipliers is a bounded line segment and a relaxed strict complementary slackness (SCS) condition holds. A new constraint qualification is presented in this paper: We assume that the MFCQ holds and the active constraint rank deficiency is 1. This assumption relaxes the linear independence constraint qualification and enforces the MFCQ. In addition, the set of Lagrange multipliers is a bounded line segment.