ABSTRACT The study of optimality conditions and constraint qualification is a key topic in nonlinear optimization. In this work, we present a reformulation of the well-known second-order constraint qualification described by McCormick in 17. This reformulation is based on the use of feasible arcs, but is independent of Lagrange multipliers. Using such a reformulation, we can show that a local minimizer verifies the strong second-order necessary optimality condition. We can also prove that the reformulation is weaker than the known relaxed constant rank constraint qualification in 19. Furthermore, we demonstrate that the condition is neither related to the MFCQ + WCR in 8 nor to the CCP2 condition, the companion constraint qualification associated with the second-order sequential optimality condition AKKT 2 in 5.

In this paper, two different approaches to solve underdetermined nonlinear system of equations are proposed. In one of them, the derivative-free method defined by La Cruz, Martínez and Raydan for solving square nonlinear systems is modified and extended to cope with the underdetermined case. The other approach is a Quasi-Newton method that uses the Broyden update formula and the globalized line search that combines the strategy of Grippo, Lampariello and Lucidi with the Li and Fukushima one. Global convergence results for both methods are proved and numerical experiments are presented.