Moment-based distributionally robust optimization (DRO) provides an
optimization framework to integrate statistical information with traditional
optimization approaches. Under this framework, one assumes that the underlying
joint distribution of random parameters runs in a distributional ambiguity set
constructed by moment information and makes decisions against the worst-case
distribution within the set. Although most moment-based DRO problems can be
reformulated as semidefinite programming (SDP) problems that can be solved in
polynomial time, solving high-dimensional SDPs is still time-consuming. Unlike
existing approximation approaches that first reduce the dimensionality of
random parameters and then solve the approximated SDPs, we propose an optimized
dimensionality reduction (ODR) approach. We first show that the ranks of the
matrices in the SDP reformulations are small, by which we are then motivated to
integrate the dimensionality reduction of random parameters with the subsequent
optimization problems. Such integration enables two outer and one inner
approximations of the original problem, all of which are low-dimensional SDPs
that can be solved efficiently. More importantly, these approximations can
theoretically achieve the optimal value of the original high-dimensional SDPs.
As these approximations are nonconvex SDPs, we develop modified Alternating
Direction Method of Multipliers (ADMM) algorithms to solve them efficiently. We
demonstrate the effectiveness of our proposed ODR approach and algorithm in
solving two practical problems. Numerical results show significant advantages
of our approach on the computational time and solution quality over the three
best possible benchmark approaches. Our approach can obtain an optimal or
near-optimal (mostly within 0.1%) solution and reduce the computational time by
up to three orders of magnitude