This paper investigates a portfolio optimization problem under uncertainty on the stock returns, where the manager seeks to achieve an appropriate trade-off between the expected portfolio return and the risk of loss. The uncertainty set consists of a finite set of scenarios occurring with equal probability. We introduce a new robustness criterion, called pw-robustness, which seeks to maximize the portfolio return in a proportion p of scenarios and guarantees a minimum return over all scenarios. We model this optimization problem as a mixed-integer programming problem. Through extensive numerical experiments, we identify the instances that can be solved to optimality in an acceptable time using off-the-shelf software. For the instances that cannot be solved to optimality within the time frame, we propose and test a heuristic that exhibits excellent practical performance in terms of computation time and solution quality for the problems we consider. This new criterion and our heuristic methods therefore exhibit great promise to tackle robustness problems when the uncertainty set consists of a large number of scenarios.
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We would like to dedicate our work to Professor Bernard Roy (1934–2017) who passed away as we were revising this paper. Professor Roy, who founded the LAMSADE group (Laboratoire d’Analyse et de Modélisation des Systèmes pour l’Aide à la Décision), was instrumental in getting us started in this area of research and provided valuable feedback on our paper. We are deeply honored to have benefited from the guidance of this pioneer of operations research. We would also like to thank two anonymous referees, whose comments have significantly improved the paper.
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