Using Semidefinite Programming to Minimize Polynomials

Optimization problems arise in widely varying contexts. The general optimization problem is to find the minimum value of a certain function, the objective, on a certain set, defined by constraints. To make such problems amenable to analysis, further restrictions must be imposed on the kinds of objectives and constraints that may arise. A priori, it might seem useful to require them to be polynomial. After all, the entire toolbox of algebra and calculus is available to deal with polynomials. They can be represented, evaluated, and manipulated easily, and they are often used to approximate more complicated functions. Partly for these reasons, they arise in many kinds of applications. As this course has demonstrated, it so happens that this is not the most fruitful restriction for optimization problems. Even problems with quadratic objectives and quadratic constraints may be difficult to solve. Rather, it is the field of convex optimization which has developed a body of theory and practice leading to computationally effective solution procedures. However, the aforementioned reasons why polynomial optimization would be desirable are still valid. Thus, 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.