Adaptive robust optimization with applications in inventory and revenue management

Abstract

In this thesis, we examine a recent paradigm for solving dynamic optimization problems under uncertainty, whereby one considers decisions that depend directly on the sequence of observed disturbances. The resulting policies, called recourse decision rules, originated in Stochastic Programming, and have been widely adopted in recent works in Robust Control and Robust Optimization; the specific subclass of affine policies has been found to be tractable and to deliver excellent empirical performance in several relevant models and applications. In the first chapter of the thesis, using ideas from polyhedral geometry, we prove that disturbance-affine policies are optimal in the context of a one-dimensional, constrained dynamical system. Our approach leads to policies that can be computed by solving a single linear program, and which bear an interesting decomposition property, which we explore in connection with a classical inventory management problem. The result also underscores a fundamental distinction between robust and stochastic models for dynamic optimization, with the former resulting in qualitatively simpler problems than the latter. In the second chapter, we introduce a hierarchy of polynomial policies that are also directly parameterized in the observed uncertainties, and that can be efficiently computed using semidefinite optimization methods. The hierarchy is asymptotically optimal and guaranteed to improve over affine policies for a large class of relevant problems. To test our framework, we consider two problem instances arising in inventory management, for which we find that quadratic policies considerably improve over affine ones, while cubic policies essentially close the optimality gap. In the final chapter, we examine the problem of dynamically pricing inventories in multiple items, in order to maximize revenues. For a linear demand function, we propose a distributionally robust uncertainty model, argue how it can be constructed from limited historical data, and show how pricing policies depending on the observed model mis-specifications can be computed by solving second-order conic or semidefinite optimization problems. We calibrate and test our model using both synthetic data, as well as real data from a large US retailer. Extensive Monte-Carlo simulations show 3 that adaptive robust policies considerably improve over open-loop formulations, and are competitive with popular heuristics in the literature.