  The difficulty in exploring potential energy surfaces, which are nonconvex,
stems from the presence of many local minima, typically separated by high
barriers and often disconnected in configurational space. We obtain the global
minimum on model potential energy surfaces without sampling any minima a
priori. Instead, a different problem is derived, which is convex and hence easy
to solve, but which is guaranteed to either have the same solution or to be a
lower bound to the true solution. A systematic way for improving the latter
solutions is also given. Because many nonconvex problems are projections of
higher dimensional convex problems, Parrilo has recently showed that by
obtaining a sum of squares decomposition of the original problem, which can be
subsequently transformed to a semidefinite programme a large class of
non-convex problems can be solved efficiently. The semidefinite duality
formulation also provides a proof that the global minimum of the energy surface
has either been found exactly or has been bounded from below. It additionally
provides physical insight into the problem through a geometric interpretation.
The sum of squares polynomial representation of the potential energy surface
may further reveal information about the nature of the potential energy
surface. We demonstrate the applicability of this approach to low dimensional
potential energy landscapes and discuss its merits and shortcomings.
