Abstract

In this work we present new weak conditions that ensure the valid- ity of necessary second-order optimality conditions (SOC) for nonlinear optimization. We are able to prove that weak and strong SOCs hold for all Lagrange multipliers using Abadie-type assumptions. We also prove weak and strong SOCs for at least one Lagrange multiplier imposing the Mangasarian-Fromovitz constraint qualification and a weak constant rank assumption.