Globalized distributionally robust optimization problems under the moment-based framework

This paper is devoted to reduce the conservativeness of distributionally robust optimization with moments information. Since the optimal solution of distributionally robust optimization is required to be feasible for all uncertain distributions in a given ambiguity distribution set and so the conservativeness of the optimal solution is inevitable. To address this issue, we introduce the globalized distributionally robust counterpart (GDRC) which allows constraint violations controlled by functional distance of the true distribution to the inner distribution set. We obtain the deterministic equivalent forms for several GDRCs under the moment-based framework. To be specific, we show the deterministic equivalent systems of inequalities for GDRCs under second order moment information with a separable convex distance function and a special jointly convex function, respectively. We also obtain the deterministic equivalent inequality for GDRC under first order moment and support information. The computationally tractable examples are presented for these GDRCs. Numerical tests of a portfolio optimization problem are given to show the effectiveness of our method and the results demonstrate that the globalized distributionally robust solution is non-conservative and flexible compared to the distributionally robust solution.