Achieving analogous convex lower-level results with simple constraint sets (balls or boxes)

Establish representation results analogous to the convex-compact semi-algebraic case—namely, characterizing the full class of semi-algebraic lower (or upper) semicontinuous functions bounded on compacts as value functions—when the lower-level feasible set is restricted to simple convex compact sets such as Euclidean balls or boxes, under a convex lower-level objective.

Background

The paper proves equality results for the classes of value functions obtainable with convex lower-level objectives over general convex, compact, semi-algebraic feasible sets, but for box constraints with convex lower-level objectives it only establishes inclusion for piecewise polynomials.

The authors explicitly leave open whether analogous tight results can be achieved with simpler constraint sets like balls or boxes, which would be of practical interest given their prevalence in optimization models.