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

AbstractLet f,gi,i=1,…,l,hj,j=1,…,m, be polynomials on Rn and S≔{x∈Rn∣gi(x)=0,i=1,…,l,hj(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