Explicit polynomial representation of the convex, compact, semi-algebraic feasible set arising from monomial-lifting convexification

Develop an explicit and efficient representation using polynomial equalities and inequalities for the convex, compact, semi-algebraic lower-level feasible set constructed via the monomial embedding and convex-hull lifting in the proof of the equality characterization for convex, compact, semi-algebraic constraints (i.e., represent Conv{M_d(𝒴)} by explicit polynomial constraints).

Background

To obtain equality characterizations for value functions under convex, compact, semi-algebraic lower-level constraints, the authors reformulate polynomial optimization via a monomial map M_d and take the convex hull of its image, producing a convex, compact, semi-algebraic set used as the lower-level feasible set.

They highlight that, unlike earlier constructions, this feasible set lacks an explicit, efficient polynomial description, which is important for practical polynomial optimization and related semidefinite relaxations.