In this paper we survey the primary research, both theoretical and applied, in the area of robust optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multistage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.

/pdf/theory-and-applications-of-robust-optimization-3tpe0fjexc.pdf

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.

In multistage problems, decisions are implemented sequentially, and thus may depend on past realizations of the uncertainty. Examples of such problems abound in applications of stochastic control and operations research; yet, where robust optimization has made great progress in providing a tractable formulation for a broad class of single-stage optimization problems with uncertainty, multistage problems present significant tractability challenges. In this paper we consider an adaptability model designed with discrete second stage variables in mind. We propose a hierarchy of increasing adaptability that bridges the gap between the static robust formulation, and the fully adaptable formulation. We study the geometry, complexity, formulations, algorithms, examples and computational results for finite adaptability. In contrast to the model of affine adaptability proposed in, our proposed framework can accommodate discrete variables. In terms of performance for continuous linear optimization, the two frameworks are complementary, in the sense that we provide examples that the proposed framework provides stronger solutions and vice versa. We prove a positive tractability result in the regime where we expect finite adaptability to perform well, and illustrate this claim with an application to Air Traffic Control.

/pdf/finite-adaptability-in-multistage-linear-optimization-4jljgh9s3f.pdf

Finite Adaptability in Multistage Linear Optimization

In recent years, decision rules have been established as the preferred solution method for addressing computationally demanding, multistage adaptive optimization problems. Despite their success, existing decision rules (a) are typically constrained by their a priori design and (b) do not incorporate in their modeling adaptive binary decisions. To address these problems, we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions, respectively. We then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed-integer optimization. We demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems. We provide global lower bounds and show that our approach is (i) practically tractable and (ii) provides high quality solutions that outperform alternative methods.

/pdf/design-of-near-optimal-decision-rules-in-multistage-adaptive-4k03y5zywa.pdf

Design of Near Optimal Decision Rules in Multistage Adaptive Mixed-Integer Optimization

Decision rules provide a flexible toolbox for solving computationally demanding, multistage adaptive optimization problems. There is a plethora of real-valued decision rules that are highly scalable and achieve good quality solutions. On the other hand, existing binary decision rule structures tend to produce good quality solutions at the expense of limited scalability and are typically confined to worst-case optimization problems. To address these issues, we first propose a linearly parameterised binary decision rule structure and derive the exact reformulation of the decision rule problem. In the cases where the resulting optimization problem grows exponentially with respect to the problem data, we provide a systematic methodology that trades-off scalability and optimality, resulting to practical binary decision rules. We also apply the proposed binary decision rules to the class of problems with random-recourse and show that they share similar complexity as the fixed-recourse problems. Our numerical results demonstrate the effectiveness of the proposed binary decision rules and show that they are (i) highly scalable and (ii) provide high quality solutions.

/pdf/binary-decision-rules-for-multistage-adaptive-mixed-integer-4vhpjcsokp.pdf

Binary decision rules for multistage adaptive mixed-integer optimization

Optimization under uncertainty is a central ingredient for analyzing and designing systems with incomplete information. This thesis addresses uncertainty in optimization, in a dynamic framework where information is revealed sequentially, and future decisions are adaptable, i.e., they depend functionally on the information revealed in the past. Such problems arise in applications where actions are repeated over a time horizon (e.g., portfolio management, or dynamic scheduling problems), or that have multiple planning stages (e.g., network design). The first part of the thesis focuses on the robust optimization approach to systems with uncertainty. Unlike the probability-driven stochastic programming approach, robust optimization is built on deterministic set-based formulations of uncertainty. This thesis seeks to place Robust Optimization within a dynamic framework. In particular, we introduce the notion of finite adaptability. Using geometric results, we characterize the benefits of adaptability, and use these theoretical results to design efficient algorithms for finding near-optimal protocols. Among the novel contributions of the work are the capacity to accommodate discrete variables, and the development of a hierarchy of adaptability. The second part of the thesis takes a data-driven view to uncertainty. The central questions are (a) how can we construct adaptability in multi-stage optimization problems given only data, and (b) what feasibility guarantees can we provide. Multistage Stochastic Optimization typically requires exponentially many data points. Robust Optimization, on the other hand, has a very limited ability to address multi-stage optimization in an adaptable manner. We present a hybrid sample-based robust optimization methodology for constructing adaptability in multi-stage optimization problems, that is both tractable and also flexible, offering a hierarchy of adaptability. We prove polynomial upper bounds on sample complexity. We further extend our results to multi-stage problems with integer variables in the future stages. We illustrate the ideas above on several problems in Network Design, and Portfolio Optimization. The last part of the thesis focuses on an application of adaptability, in particular, the ideas of finite adaptability from the first part of the thesis, to the problem of air traffic control. The main problem is to sequentially schedule the departures, routes, ground-holding, and air-holding, for every flight over the national air space (NAS). The schedule seeks to minimize the aggregate delay incurred, while satisfying capacity constraints that specify the maximum number of flights that can take off or land at a particular airport, or fly over the same sector of the NAS at any given time. These capacities are impacted by the weather conditions. Since we receive an initial weather forecast, and then updates throughout the day, we naturally have a multistage optimization problem, with sequentially revealed uncertainty. We show that finite adaptability is natural, since the scheduling problem is inherently finite, and furthermore the uncertainty set is low-dimensional. We illustrate both the applicability of finite adaptability, and also its effectiveness, through several examples. Thesis Supervisor: Dimitris Bertsimas Title: Boeing Professor of Operations Research

Constantine Caramanis

Papers

Adaptable optimization : theory and algorithms