Strictness of the inclusion between convex lower-level and general polynomial bilevel value-function classes

Determine whether, for each modeling mode (optimistic or pessimistic) and box type (bounded or unbounded), the class of value functions representable by polynomial bilevel problems with convex lower-level objectives is strictly smaller than, or equal to, the corresponding class of value functions representable by general polynomial bilevel problems without convexity restrictions. Concretely, decide the strictness or equality of the inclusion between the convex-lower-level value-function class and the general polynomial bilevel value-function class under the same constraint type.

Background

The paper defines two families of value-function classes for polynomial bilevel programs: one allowing arbitrary (nonconvex) lower-level objectives, and one restricting the lower level to be convex. For each of these, the authors consider optimistic and pessimistic modeling modes as well as bounded or unbounded box constraints for the lower-level feasible set.

They note the obvious inclusion of the convex-lower-level class within the general class but state that it is unknown whether this inclusion is strict. This question bears on how much convexity at the lower level reduces the expressive power of polynomial bilevel value functions.