Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Abstract

Let 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 .