Polynomial Optimization via Asymptotic Critical Values
Certifying whether a semi-algebraic function assumes only nonnegative values is an important problem in real algebraic geometry, having applications to other areas of mathematics, such as optimization.
An equivalent problem asks for approximating the infimum of a polynomial function ƒ : X → Rn over a semi-algebraic set X ⊆ Rn.
If the infimum exists and is not attained, it partakes in one of several points in R, in the vicinity of which the topology of the fibers under ƒ changes.
These points are called the ″asymptotic critical values″ of ƒ, and can be computed effectively under some strict assumptions on X.
In this talk, I will present the state of the art around computing and approximating the asymptotic critical values of complex and real polynomial functions.
I will also introduce a new algorithm for performing this task under some mild non-degeneracy assumptions on the semi-algebraic set X.