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An algorithm for semi-infinite polynomial optimization

We consider the semi-infinite optimization problem: $f^*:=\min_{x\in X}\:\{f(x): g(x,y)\,\leq \,0,\:\forally\in Y_x\}$, where $f,g$ are polynomials and $X\subset R^n$ as well as $Y_\x\subset R^p$, $x\in X$, are compact basic semi-algebraic sets. To approximate $f^*$ we proceed in two steps. First, we use the "joint+marginal" approach of the author to approximate from above the function $x\mapstoΦ(x)=\sup \{g(x,y): y\in Y_x\}$ by a polynomial $Φ_d\geqΦ$, of degree at most $2d$, with the strong property that $Φ_d$ converges to $Φ$ for the $L_1$-norm, as $d\to\infty$ (and in particular, almost uniformly for some subsequence $(d_\ell)$, $\ell\in\N$). Then we solve the polynomial optimization problem $f^*_d=\min_{x\in X} \{f(x): Φ_d(x)\leq0\}$ via a (by now standard) hierarchy of semidefinite relaxations. It turns out that the optimal value $f^*_d\geq f^*$ converges to $f^*$ as $d\to\infty$. In practice we let $d$ be fixed, small, and relax the constraint $Φ_d\leq0$ to $Φ_d(x)\leqε$ with $ε>0$, allowing to change $ε$ dynamically.