Tightness of inclusions for box-constrained convex lower-level representations

Ascertain the tightness of the inclusions established for box-constrained convex lower-level polynomial bilevel programs—specifically, whether the inclusions relating piecewise polynomial functions and semi-algebraic (and semicontinuous/bounded-on-compacts) value-function classes are equalities or strict inclusions—by proving equality or constructing counterexamples.

Background

Earlier in the section, the authors show that with convex lower-level objectives and box constraints, the value-function class includes all piecewise polynomial functions and is contained within semi-algebraic classes with appropriate semicontinuity and boundedness properties. They do not establish equality of these inclusions.

They explicitly state that the tightness of these inclusions for box-constrained convex lower-level problems remains an open question, and proceed to paper a related setting with arbitrary convex, compact, semi-algebraic constraint sets.