  This paper is devoted to reduce the conservatism 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
conservatism 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 uncertainty distribution set. We obtain the
deterministic equivalent forms for several GDRCs under the moment-based
framework. To be specific, we show the deterministic equivalent system of
inequalities for the GDRCs under second order moment information with a
separable distance function and a jointly convex distance function,
respectively. Moreover, the feasible set of the system is convex. We also
develop the deterministic equivalent inequality for the GDRC under first order
moment and support information. The computationally tractable examples are
presented for these GDRCs. A numerical tests of a portfolio optimization
problem is given to show the efficiency of our methods and the results
demonstrate that the globalized distributionally robust solutions is
non-conservative and flexible compared to the distributionally robust
solutions.
