  Optimization problems with discrete decisions are nonconvex and thus lack
strong duality, which limits the usefulness of tools such as shadow prices and
the KKT conditions. It was shown in Burer(2009) that mixed-binary quadratic
programs can be written as completely positive programs, which are convex.
Completely positive reformulations of discrete optimization problems therefore
have strong duality if a constraint qualification is satisfied. We apply this
perspective in two ways. First, we write unit commitment in power systems as a
completely positive program, and use the dual copositive program to design a
new pricing mechanism. Second, we reformulate integer programming games in
terms of completely positive programming, and use the KKT conditions to solve
for pure strategy Nash equilibria. To facilitate implementation, we also design
a cutting plane algorithm for solving copositive programs exactly.
