Global Optimization of Polynomials over Real Algebraic Sets

Let f , g 1 ,..., g s be polynomials in R[ X 1 ,..., X n ]. Based on topological properties of generalized critical values, the authors propose a method to compute the global infimum f * of f over an arbitrary given real algebraic set V = { x ∈ R n | g 1 ( x ) = 0,..., g s ( x ) = 0}, where V is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing f over a basic closed semi-algebraic set S = { x ∈ R n | g 1 ( x ) ≥ 0,..., g s ( x ) ≥ 0}.