Adaptive Distributionally Robust Optimization

Abstract

We develop a modular and tractable framework for solving an adaptive distributionally robust linear optimization problem, where we minimize the worst-case expected cost over an ambiguity set of probability distributions. The adaptive distrbutaionally robust optimization framework caters for dynamic decision making, where decisions adapt to the uncertain outcomes as they unfold in stages. For tractability considerations, we focus on a class of second-order conic (SOC) representable ambiguity set, though our results can easily be extended to more general conic representations. We show that the adaptive distributionally robust linear optimization problem can be formulated as a classical robust optimization problem. To obtain a tractable formulation, we approximate the adaptive distributionally robust optimization problem using linear decision rule (LDR) techniques. More interestingly, by incorporating the primary and auxiliary random variables of the lifted ambiguity set in the LDR approximation, we can significantly improve the solutions, and for a class of adaptive distributionally robust optimization problems, exact solutions can also be obtained. Using the new LDR approximation, we can transform the distributionally adaptive robust optimization problem to a classical robust optimization problem with an SOC representable uncertainty set. 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 medical appointment scheduling and inventory management problems.

On distributionally robust chance constrained programs with Wasserstein distance

“…Distributionally robust optimization (DRO) has been an alternative modeling paradigm for optimization under uncertainty, where the probability distributions of random parameters are not fully known. Interested readers are referred to Rahimian and Mehrotra (2019) (i) Moment ambiguity set is specified by the acquired knowledge of some moments (e.g., known first two moments), and has been successfully applied to many different settings (see for example, Delage and Ye 2010, Bertsimas et al 2010, Goh and Sim 2010, Bertsimas et al 2018b, Wiesemann et al 2014, Hanasusanto et al 2015, Natarajan and Teo 2017, Li et al 2017, Xie and Ahmed 2018a,b, Zhang et al 2018b). Delage and Ye (2010) shows that if the first two moments are known or bounded from above, and the recourse function can be expressed as piecewise maximum of a finite number of functions which are convex in x and concave in the random parametersξ, then the function Z(x) have a tractable representation.…”

Robust stochastic optimization with convex risk measures: A discretized subgradient scheme

“…Thus, the ambiguity set (17) includes (11)-(15) as special cases. A computationally tractable case whereH(ω) is affine andḠ i (ω) is cone-representable has been considered by Bertsimas et al [8].…”

Optimization under uncertainty in the era of big data and deep learning: When machine learning meets mathematical programming

“…To account for the sequential decision-making process, researchers recently developed the adaptive DRO method by incorporating recourse decision variables [123,124]. A general twostage data-driven stochastic programming model is presented in the following form:…”