As I understand polynomial optimization on a domain is based on the use of Putinar’s Positivestellensatz. Does this imply that the domain needs to be compact? Or am I misunderstanding something? The SumOfSquares manuals do not mention the compactness requirement.
As best I can understand it, no. Baldi and Slot, Nie and Schweighofer
Putinar’s Positivstellensatz requires the quadratic module generated by the domain-defining polynomials to be Archimedean, not the domain to be explicitly compact.The Archimedean condition means there exists R > 0 such that R - ||x||^2 belongs to the quadratic module, which implies the semi-algebraic set S(g) = {x | g_i(x) >= 0} is compact. Putinar enforces nonnegativity on domains like @set x >= 0 && y >= 0 && x + y <= 1 , which is Archimedean.
However, the gotcha may lurk in that non-Archimedean (non-compact) domains require alternative certificates like Helton-Nie or Laurent, not standard Putinar; manuals omit details as users must ensure the condition.
Salt disclaimer. It has been almost 60 years since I’ve taken a math course.
Thank you for the answer. (1) It seems to me that this SHOULD be mentioned in the manuals. (2) As far as I can understand the Archemedean property implies compactness, so the domain DOES need to be compact. Or did I misunderstand something? I also have not been to a math course in 55 years, Putinar’s positivesellensatz is more recent than that.
If it has the Archemedean, then it is compact, otherwise Putinar is out of scope. Again, based on my necessarily infirm grasp of the subject.