Extension of the convex-hull reformulation to unbounded lower-level constraints

Develop a method that extends the convex-hull-based reformulation used to obtain equality results for convex, compact, semi-algebraic lower-level feasible sets to the case where the feasible set is convex, closed, semi-algebraic, and unbounded, overcoming the failure of closedness of the convex hull in the unbounded setting.

Background

The convex reformulation relies on replacing the lower-level polynomial optimization by a convex program over the convex hull of the monomial map image of the feasible set, which assumes compactness to preserve closedness.

For unbounded convex, closed, semi-algebraic feasible sets, the authors explain that the same convex-hull idea fails because the convex hull of a closed set may not be closed, and they did not find a workaround.