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.

Theory and Applications of Robust Optimization

JuMP is an open-source modeling language that allows users to express a wide range of optimization problems (linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear) in a high-level, algebraic syntax. JuMP takes advantage of advanced features of the Julia programming language to offer unique functionality while achieving performance on par with commercial modeling tools for standard tasks. In this work we will provide benchmarks, present the novel aspects of the implementation, and discuss how JuMP can be extended to new problem classes and composed with state-of-the-art tools for visualization and interactivity.

JuMP: A Modeling Language for Mathematical Optimization

Distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty. The distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker's prior information. In this paper, we propose a unifying framework for modeling and solving distributionally robust optimization problems. We introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold. These ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases. They also allow us to characterize distributional families in terms of several classical and/or robust statistical indicators that have not yet been studied in the context of robust optimization. We determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable. We also provide tractable conservative approximations for problems that violate these conditions.

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Distributionally Robust Convex Optimization

In this paper we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust and more flexible than the standard technique of using linear rules. Our framework begins by first affinely extending the set of primitive uncertainties to generate new linear decision rules of larger dimensions and is therefore more flexible. Next, we develop new piecewise-linear decision rules that allow a more flexible reformulation of the original problem. The reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables. Finally, we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations. These bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds and next integrating them into a combined bound that is better than each of the individual bounds.

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Distributionally Robust Optimization and Its Tractable Approximations

Renewable energy resources such as wind and solar are an important component of a microgrid. However, the inherent intermittency and variability of such resources complicates microgrid operations. Meanwhile, more controllable loads (e.g., plug-in electric vehicles), distributed generators (e.g., micro gas turbines and diesel generators), and distributed energy storage devices (e.g., battery banks) are being integrated into the microgrid operation. To address the operational challenges associated with these technologies and energy resources, this paper formulates a stochastic problem for microgrid energy scheduling. The proposed problem formulation minimizes the expected operational cost of the microgrid and power losses while accommodating the intermittent nature of renewable energy resources. Case studies are performed on a modified IEEE 37-bus test feeder. The simulation results demonstrate the effectiveness and accuracy of the proposed stochastic microgrid energy scheduling model.

Stochastic Energy Scheduling in Microgrids With Intermittent Renewable Energy Resources

Introduction to stochastic programming

We present a framework for solving the strategic problem of timing the location of
facilities and the assigning of customers to facilities in a multi-period environment under
uncertainty in the timing at which the customers need to be assigned to the facilities and
the minimum number of customers to be served at each time horizon. By considering
a compact mathematical representation of the Deterministic Equivalent Model, we specialize
the so-called Branch-and-Fix Coordination algorithmic frame-work. It exploits the
structure of the model and, specically, the non-anticipativity constraints for the variables.
The algorithm uses the Twin Node Family concept and the Lagrangean substitution
bounding. Our procedure is specically designed for coordinating the selection of the branching
TNF, such that the nonanticipativity constraints are satised. Some computational
experience is reported.

María Celeste Pizarro Romero

Papers

Introduction to stochastic programming

On solving the multi-period location-assignment problem under uncertainty