Regularized distributionally robust optimization with application to the index tracking problem

In recent years, distributionally robust optimization (DRO) has received a lot of interest due to its ability to reduce the worst-case risk when there is a perturbation to the data-generating distribution. A good statistical model is expected to perform well on both the normal and the perturbed data. On the other hand, variable selection and regularization is a research area that aims to identify the important features and remove the redundant ones. It helps to improve the prediction accuracy as well as the interpretability of the model. In this paper, we propose an optimization model that is a regularized version of the canonical distributionally robust optimization (DRO) problem where the ambiguity set is described by a general class of divergence measures that admit a suitable conic structure. The divergence measures we examined include several popular divergence measures used in the literature such as the Kullback–Leibler divergence, total variation, and the Chi-divergence. By exploiting the conic representability of the divergence measure, we show that the regularized DRO problem can be equivalently reformulated as a nonlinear conic programming problem. In the case where the regularization is convex and semi-definite programming representable, the reformulation can be further simplified as a tractable linear conic program and hence can be efficiently solved via existing software. More generally, if the regularization can be written as a difference of convex functions, we demonstrate that a solution for the regularized DRO problem can be found by solving a sequence of conic linear programming problems. Finally, we apply the proposed regularized DRO model to both simulated and real financial data and demonstrate its superior performance in comparison with some non-robust models.