  Farkas' lemma is a fundamental result from linear programming providing
linear certificates for infeasibility of systems of linear inequalities. In
semidefinite programming, such linear certificates only exist for strongly
infeasible linear matrix inequalities. We provide nonlinear algebraic
certificates for all infeasible linear matrix inequalities in the spirit of
real algebraic geometry: A linear matrix inequality is infeasible if and only
if -1 lies in the quadratic module associated to it. We also present a new
exact duality theory for semidefinite programming, motivated by the real
radical and sums of squares certificates from real algebraic geometry.
