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

Let f,gi,i=1,…,l,hj,j=1,…,mf,gi,i=1,…,l,hj,j=1,…,m, be polynomials on RnRn and S≔{x∈Rn∣gi(x)=0,i=1,…,l,hj(x)≥0,j=1,…,m}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 ff on the semialgebraic set SS via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial ff does not attain its infimum on SS. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of ff on SS and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of ff 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 ff on SS.