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

Determine whether, for each mode in {optimistic, pessimistic} and each box type in {bounded, unbounded}, the inclusion C^{mode}_{boxtype} ⊆ P^{mode}_{boxtype} is strict or not, where P^{mode}_{boxtype} denotes the class of value functions realizable by polynomial bilevel problems with the specified mode and box constraints and no convexity requirement on the lower-level objective, and C^{mode}_{boxtype} denotes the corresponding class when the lower-level objective is convex in the lower-level variable.

Background

The paper defines P^{mode}_{boxtype} as the set of value functions arising from polynomial bilevel problems with specified optimistic or pessimistic mode and a lower-level feasible set that is a box (bounded or unbounded), without imposing convexity on the lower-level objective. The class C^{mode}_{boxtype} is defined similarly but requires the lower-level objective to be convex in the lower-level variable.

By construction, imposing convexity on the lower-level objective yields C^{mode}_{boxtype} ⊆ P^{mode}_{boxtype}. The authors show large expressivity for C^{mode}_{boxtype} (e.g., inclusion of piecewise polynomial functions) but explicitly note that they do not know whether the inclusion is strict.