Summary: We study the convergence of a sequential quadratic programming algorithm for the nonlinear programming problem. The Hessian of the quadratic program is the sum of an approximation of the Lagrangian and of a multiple of the identity that allows us to penalize the displacement. Assuming only that the direction is a stationary point of the current quadratic program we study the local convergence properties without strict complementarity. In particular, we use a very weak condition on the approximation of the Hessian to the Lagrangian. We obtain some global and superlinearly convergent algorithm under weak hypotheses. As a particular case we formulate an extension of Newton's method that is quadratically convergent to a point satisfying a strong sufficient second-order condition.