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a b s t r a c tLet f , g i , i = 1, . . . , l, h j , j = 1, . . . , m, be polynomials on R n and S := {x ∈ R n | g i (x) = 0, i = 1, . . . , l, h j (x) ≥ 0, j = 1, . . . , m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.

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