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- W4220774658 abstract "We provide a new degree bound on the weighted sum-of-squares (SOS) polynomials for Putinar–Vasilescu's Positivstellensatz. This leads to another Positivstellensatz saying that if f is a polynomial of degree at most 2df nonnegative on a semialgebraic set having nonempty interior defined by finitely many polynomial inequalities gj(x)≥0, j=1,…,m with g1:=L−‖x‖22 for some L>0, then there exist positive constants c¯ and c depending on f,gj such that for any ε>0, for all k≥c¯ε−c, f has the decomposition(1)(1+‖x‖22)k(f+ε)=σ0+∑j=1mσjgj, for some SOS polynomials σj being such that the degrees of σ0,σjgj are at most 2(df+k). Here ‖⋅‖2 denotes the ℓ2 vector norm. As a consequence, we obtain a converging hierarchy of semidefinite relaxations for lower bounds in polynomial optimization on basic compact semialgebraic sets. The complexity of this hierarchy is O(ε−c) for prescribed accuracy ε>0. In particular, if m=L=1 then c=65, yielding the complexity O(ε−65) for the minimization of a polynomial on the unit ball." @default.
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- W4220774658 date "2022-10-01" @default.
- W4220774658 modified "2023-10-01" @default.
- W4220774658 title "On the complexity of Putinar–Vasilescu's Positivstellensatz" @default.
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- W4220774658 doi "https://doi.org/10.1016/j.jco.2022.101663" @default.
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