Using Semidefinite Programming to Minimize Polynomials (original) (raw)

this paper attempts to explain and demonstrate how to use the techniques of convex optimization to approximately (often, exactly) solve polynomial optimization problems. For concreteness, the problems will be posed as minimization problems. For simplicity, the constraints will be linear, or absent. 1 2 Why Not Just Calculus? At first it might not be clear why this problem is nontrivial, particularly since a simple domain has been assumed. After all, everyone learns in calculus that a function attains its minimum (if it has one) either at a critical point or on the boundary of its domain. Since the function in question is polynomial, its partial derivatives are easy to compute. So, just set them all to zero and evaluate the function at each of those points. If it is in doubt whether or not a particular point is a minimum, evaluate the Hessian at that point. This requires computing second derivatives, which is also easy. To find the minimum on the boundary, simply specialize the polyn...