An algorithm for semi‐infinite polynomial optimization

We consider the semi‐infinite optimization problem: $f^{*}≔{\min }_{x\in X}\{f(x):\text{g}(x\text{,}y)\le 0\text{,}\forall y\in {\text{Y}}_x\}$ via a (by now standard) hierarchy of semidefinite relaxations, and for increasing values of d. In practice d is fixed, small, and one relaxes the constraint θ d ≤0 to θ d (x)≤ϵ with ϵ>0, allowing us to change ϵ dynamically. As d increases, the limit of the optimal value $f_d^{ϵ}$ is bounded above by f ^*+ϵ.