In many applications of mathematics, including areas such as financial mathematics and control theory, optimization plays an important role.
Often the criterion function is a polynomial or a rational function of the unknowns, which we will take to be real numbers here. In such a case the first order conditions for optimality can be written in the form of a polynomial system of equations and we
speak of an algebraic optimization problem. The solutions of the polynomial system are called the critical points of the criterion function. The corresponding values of the criterion function are called critical values. There are only a finite number of such critical values and one can construct a univariate nonzero polynomial which is zero on the critical value set. Such a polynomial will be called a critical value polynomial
(CVP).
In the talk we will explain methods to obtain such a critical value polynomial and some generalizations and their usage for determining the (global!) optimal value of
the criterion.