We develop a general framework for selecting a small pool of candidate solutions to maximize the chances that one will be optimal for a combinatorial optimization problem, under a linear and additive random payoff function. We formulate this problem using a two-stage distributionally robust model, with a mixed 0–1 semidefinite program. This approach allows us to exploit the “diversification” effect inherent in the problem to address how different candidate solutions can be selected to improve the chances that one will attain a high ex post payoff. More interestingly, using this distributionally robust optimization approach, our model recovers the “evil twin” strategy, well known in the field of football pool betting, under appropriate settings.
We also address the computational challenges of scaling up our approach to construct a moderate number of candidate solutions to increase the chances of finding one that performs well. To this end, we develop a sequential optimization approach based on a compact semidefinite programming reformulation of the problem. Extensive numerical results show the superiority of our approach over existing methods.
