Distributionally Adaptive Optimization

We develop a modular and tractable framework for solving a distributionally adaptive optimization problem, where we minimize the worst-case expected cost over an ambiguity set of probability distributions. The adaptive optimization framework caters for dynamic decision making, where decisions can adapt to the uncertain outcomes as they unfold in stages. We propose a second-order conic (SOC) representable ambiguity set to characterize distributional ambiguity. To obtain tractable formulation, we formulate the adaptive optimization problem using linear decision rule (LDR) approximation techniques. More interestingly, by incorporating the primary and auxiliary random variables of the lifted ambiguity set in the enhanced LDR (ELDR) approximation, we can obtain significantly better solutions and for a class of distributionally adaptive optimization problems, exact solutions can also be obtained. Using the ELDR approximation, we can transform the distributionally adaptive optimization problem to a classical robust optimization problem with an SOC representable uncertainty set. Hence, depending on the ambiguity set, the resulting framework is either a linear optimization problem or a second-order conic optimization problem (SOCP), which can be solved efficiently by general purpose commercial grade solvers. Finally, to demonstrate the potential for solving management decision problems, we develop an algebraic modeling package and illustrate how it can be used to facilitate modeling and obtain high quality solutions for addressing a medical appointment scheduling problem and a multiperiod inventory control problem.