On Sequential Quadratic Programming Methods Employing Second Derivatives

We consider sequential quadratic programming methods (SQP) globalized by linesearch for the standard exact penalty function. It is well known that if the Hessian of the Lagrangian is used in SQP subproblems, the obtained direction may not be of descent for the penalty function. The reason is that the Hessian need not be positive definite, even locally, under any natural assumptions. Thus, if a given SQP version computes the Hessian, it may need to be adjusted in some way which ensures that the computed direction becomes of descent after a finite number of Hessian modifications (for example, by consecutively adding to it some multiple of the identity matrix, or using any other technique which guarantees the needed property after a few steps). As our theoretical contribution, we show that despite the Hessian not being positive definite, such modifications are actually not needed to guarantee the descent property when the iterates are close to a solution satisfying natural assumptions. The assumptions, in fact, are exactly the same as those required for local superlinear convergence of SQP in the first place (uniqueness of the Lagrange multipliers and the second-order sufficient condition). Moreover, in our computational experiments on the Hock–Schittkowski test collection, we found (somewhat surprisingly) that with a suitable control of the penalty parameters, the Hessian modifications were actually ever needed for rather few of the problems. And even for those problems for which modifications were needed, it usually happened only on a few of the early iterations and almost never afterwards. We also compare the method with the SQPlab implementation of quasi-Newton (BFGS) SQP and with SQP using the modified Cholesky procedure, compare different rules of setting the penalty parameters, and different options of switching from the quasi-Newton to the exact Hessian version at some point along the way.