POLYNOMIAL OPTIMIZATION FROM AN ALGEBRAIC GEOMETRIC VIEWPOINT

In this talk, we discuss the application of algebraic geometry to
polynomial optimization. First, the notions of ideals, affine
varieties and Groebner bases are introduced. We show how these
notions are typically used to solve systems of multivariable
polynomial equations, using the elimination method, based on the
computation of a specific Groebner basis, or the Stetter-Möller
matrix method, which relies on the computation of eigenvalues of a
particular linear operator. We then present a technique recently
developed by Hanzon and Jibetean to link polynomial equation
solving with (global) polynomial minimization. Finally, we
investigate the algorithmic complexity of such methods and discuss
the special case dealing with polynomials
involving only two variables.