The worst-case discounted regret portfolio optimization problem

Recently a regret portfolio optimization approach is proposed by minimizing the difference between the maximum return and the sum of each portfolio return which can efficiently overcome the drawback that the classical portfolio optimization model cannot catch the core of the risk diversification. In this paper, we generalize the regret portfolio optimization approach by considering to minimize the weighted sum of the difference between the return and the sum of each portfolio return. We suppose that the decision-maker is ambiguous about the choice of weights and he choose a robust optimization approach to cope with this ambiguity. Then we aim at minimizing the maximization of the weighted sum of the difference between the return and the sum of each portfolio return, where the maximization is to be taken in all the possible distributions of the weights. We call this generalization as worst-case discounted regret portfolio optimization. In general the solution for this problem is NP hard and some approximation method is often proposed. In this paper, we suppose that the decision maker gets across some parts of information about the uncertain distributions of the weights, for example the first-order, support set and affine first-order information. By applying the duality of the semi-infinite programming, the worst-case discounted regret portfolio optimization problem with the uncertain distributions can be equivalently reformulated to a linear optimization problem, then the established solution approaches for linear optimization can all be applied to our setting. An example of a portfolio optimization problem is given to show the efficiency of our methods and the results demonstrate that our methods can satisfy the diversified property under the uncertain distributions of the weights.