Extension of the convex-hull monomial-lifting approach to unbounded lower-level constraint sets

Develop a method to extend the monomial-lifting convex-hull reformulation used for convex, compact, semi-algebraic lower-level constraint sets to unbounded lower-level feasible sets, overcoming the obstruction that the convex hull of a closed set may fail to be closed and thus invalidates the current approach.

Background

The equality results for convex lower-level problems hinge on replacing polynomial optimization over a compact semi-algebraic set with linear optimization over the convex hull of its monomial embedding. This relies on compactness and closedness to ensure appropriate convex hull properties.

The authors note that this approach does not carry over to unbounded settings because the convex hull of a closed set need not be closed, and they were unable to find a workaround.

References

Note that the same idea (if one allows $\cY$ to be an arbitrary convex, closed semi-algebraic set) does not work for unbounded cases because the convex hull of a closed set is not necessarily closed. We did not find a way around this issue, and leave this question for future work.

— Geometric and computational hardness of bilevel programming (2407.12372 - Bolte et al., 17 Jul 2024) in Section "Geometric hardness of polynomial bilevel optimization", Subsection "Polynomial bilevel problems with convex lower-level", Subsubsection "Extension to arbitrary convex, compact, semi-algebraic lower-level constraints"