Constraint deletion strategy in the inertia-controlling quadratic programming method

The inertia-controlling strategy in active set methods consists of choosing the working set so that the reduced Hessian never has more than one non-positive eigenvalue. Usually, this strategy has been implemented by permitting to delete constraints only at stationary points. This paper concerns the general inertia-controlling quadratic programming method, in which constraints may be deleted at non-stationary points. We consider the determination of the search direction when the reduced Hessian is positive definite, positive semidefinite and singular and indefinite or negative definite. Recurrence formulas are presented to update the search direction and multiplier estimates when the working set changes.