We study the choice of the Lagrange multipliers in the successive quadratic programming method (SQP) applied to the equality constrained optimization problem. It is known that the augmented Lagrangian SQP-Newton method depends on the penalty parameter only through the multiplier in the Hessian matrix of the Lagrangian function. This effectively reduces the augmented Lagrangian SQP-Newton method to the Lagrangian SQP Newton method where only the multiplier estimate depends on the penalty parameter. In this work, we construct a multiplier estimate that depends strongly on the penalty parameter and we derive a choice for the penalty parameter so that the Hessian matrix, restricted to the null space of the constraints, is positive definite and well conditioned. We demonstrate that the SQP-Newton method with this choice of Lagrange multipliers is locally and q-quadratically convergent.