Explicit polynomial description of the convex-hull-based feasible set used in the convex reformulation

Develop an explicit and computationally efficient representation, via polynomial equalities and inequalities, of the compact, convex, semi-algebraic lower-level feasible set introduced in the convex reformulation (obtained as the convex hull of the monomial map of the original variables), so that it can be directly specified within polynomial bilevel programs.

Background

To prove the equality results for value-function classes with convex lower-level objectives over convex, compact, semi-algebraic feasible sets, the authors reformulate the lower-level polynomial optimization as a convex optimization problem by mapping variables to monomials up to a given degree and taking the convex hull of the image.

While this yields the required existence of a convex, compact, semi-algebraic feasible set, the authors note that they lack an explicit and efficient polynomial description of this set, which is crucial for constructive use in polynomial optimization and for controlling dimensionality.