  Quadratically-constrained unbounded integer programs hold the distinction of
being undecidable, suggesting a possible soft-spot for Mathematical Programming
(MP) techniques, which otherwise constitute a good choice to treat integer or
mixed-integer (MI) problems. We consider the challenge of minimizing MI convex
quadratic objective functions subject to unbounded decision variables and
quadratic constraints. Given the theoretical weakness of white-box MP solvers
to handle such models, we turn to black-box meta-heuristics of the Evolution
Strategies (ESs) family, and question their capacity to solve this challenge.
Through an empirical assessment of quadratically-constrained quadratic
objective functions, across varying Hessian forms and condition numbers, we
compare the performance of the CPLEX solver to state-of-the-art MI ESs, which
handle constraints by penalty. Our systematic investigation begins where the
CPLEX solver encounters difficulties (timeouts as the search-space
dimensionality increases, (D>=30), on which we report by means of detailed
analyses. Overall, the empirical observations confirm that black-box and
white-box solvers can be competitive, especially when the constraint function
is separable, and that two common ESs' mutation operators can effectively
handle the integer unboundedness. We also conclude that conditioning and
separability are not intuitive factors in determining the complexity of this
class of MI problems, where regular versus rough landscape structures can pose
mirrored degrees of challenge for MP versus ESs.
