A robust optimization approach to online problems

Abstract

In this thesis, we consider online optimization problems that are characterized by incrementally revealed input data and sequential irrevocable decisions that must be made without complete knowledge of the future. We employ a combination of mixed integer optimization (MIO) and robust optimization (RO) methodologies in order to design new efficient online algorithms that outperform state-of-the-art methods for many important practical applications. We empirically demonstrate that RO-based algorithms are computationally tractable for instances of practical size, generate more cost-effective decisions and can simultaneously model a large class of similar online problems due to exceptional modeling power of MIO. In Part I, we consider the well-known K-server problem from the perspective of robust adaptive optimization. We propose a new tractable mixed integer linear formulation of the K-server problem that incorporates both information from the past and uncertainty about the future. By combining ideas from classical online algorithms developed in the computer science literature and robust and adaptive optimization developed in the operations research literature we propose a new method that (a) is computationally tractable, (b) almost always outperforms all other methods in numerical experiments, and (c) is stable with respect to potential errors in the assumptions about the future. In Part II, we consider several extensions of the asset-based weapon-to-target assignment problem whose objective is to protect ships in a fleet from incoming threats. We demonstrate that the new highly nonlinear MIO formulation (a) can be combined with lazy constraints techniques allowing the system designer to find optimal solutions in real time, (b) can be extended to the multi-period setting, and (c) admits a decentralized solution with limited loss of optimality. In Part III, we present a novel covariate-adaptive optimization algorithm for online allocation in clinical trials. The new approach leveraging MIO and RO techniques (a) guarantees a better between-group covariate balance in comparison with state-of- the-art methods, (b) yields statistical power at least as high as, and sometimes significantly higher than, randomization-based algorithms, and (c) is well protected against selection, investigator and accidental bias.