The problem of determining whether quadratic programming models possess either unique or multiple optimal solutions is important for empirical analyses which use a mathematical programming framework. Policy recommendations which disregard multiple optimal solutions (where they exist) are potentially incorrect and less than efficient. This paper proposes a strategy and the associated algorithm for finding all optimal solutions to any positive semidefinite linear complementarity problem. One of the main results is that the set of complementary solutions is convex. Although not obvious, this proposition is analogous to the well-known result in linear programming which states that any convex combination of optimal solutions is itself optimal.