Global Optimization of Polynomials Using the Truncated Tangency Variety and Sums of Squares

This paper proposes a method for finding the global infimum of a multivariate polynomial f via sum of squares (SOS) relaxation over its truncated tangency variety. This variety is truncated of the set of all points x ∈ Rⁿ where the level sets of f are tangent to the sphere in Rⁿ centered in the origin and with radius $\|x\|.$ It is demonstrated that: These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge monotonically, increasing to the infimum of $f.$ This opens up the possibility of solving previously intractable polynomial optimization problems.