Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem J* := max(y is an element of Omega) min(x) {f (x, y) : (x, y) is an element of Delta} where f is a polynomial and Delta subset of R(n) x R(p) as well as Omega subset of R(p) are compact basic semi- algebraic sets. We first provide a sequence of polynomial lower approximations (J(i)) subset of R [ y] of the optimal value function J (y) := minx{f (x, y) : (x, y) is an element of Delta}. The polynomial J(i) is an element of R [y] is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the " joint+ marginal" hierarchy of semidefinite relaxations associated with the parametric optimization problem y bar right arrow J (y), recently proposed in Lasserre (SIAM J Optim 20, 1995- 2022, 2010). Then for fixed i, we consider the polynomial optimization problem J(i)* := max(y){J(i) (y) : y is an element of Omega} and prove that (J) over cap (*)(i) (:= max(l=1,) (... , i) J(l)*) converges to J* as i -> infinity. Finally, for fixed l <= i, each J(l)* (and hence (J) over cap (i)*) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796-817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).