The Facet-to-Facet Property of Solutions to Convex Parametric Quadratic Programs and a new Exploration Strategy

In some of the recently-developed algorithms for convex parametric quadratic programs it is implicitly assumed that the intersection of the closures of two adjacent critical regions is a facet of both closures; this will be referred to as the facet-to-facet property. It is shown by an example, whose solution is unique, that the facet-to-facet property does not hold in general, and consequently, some existing algorithms cannot guarantee that the entire parameter space will be explored. A simple method applicable to several existing algorithms is presented for the purpose of overcoming this problem.