Tractable approximations of sets defined with quantifiers

Abstract: Given a compact basic semi-algebraic set $K\subset R^n\times R^m$, a simple set $B$ (box or ellipsoid), and some semi-algebraic function $f$, we consider sets defined with quantifiers, of the form $R_f:={x\in B: \mbox{$f(x,y)\leq 0$ for all $y$ such that $(x,y)\in K$}}$ and $D_f:={x\in B: \mbox{$f(x,y)\geq 0$ for some $y$ such that $(x,y)\in K$}}$. The former set $R_f$ is particularly useful to qualify "robust" decisions $x$ versus noise parameter $y$ (e.g. in robust optimization on some set $\mathbf{\Omega}\subset B$) whereas the latter set $D_f$ is useful (e.g. in optimization) when one does not want to work with its lifted representation ${(x,y)\in K: f(x,y)\geq 0}$. Assuming that $K_x:={y:(x,y)\in K}\neq\emptyset$ for every $x\in B$, we provide a systematic procedure to obtain a sequence of explicit inner (resp. outer) approximations that converge to $R_f$ (resp. $D_f$) in a strong sense. Another (and remarkable) feature is that each approximation is the sublevel set of a single polynomial whose vector of coefficients is an optimal solution of a semidefinite program. Several extensions are also proposed, and in particular, approximations for sets of the form $R_F:={x\in B:\mbox{$(x,y)\in F$ for all $y$ such that $(x,y)\in K$}}$, where $F$ is some other basic-semi algebraic set, and also sets defined with two quantifiers.