Robust Optimization of Large-Scale Systems

TLDR

Mathematical programming models frequently contend with noisy, erroneous, or incomplete data, which are traditionally handled either reactively through sensitivity analysis or proactively via stochastic programming. This paper characterizes desirable properties of solutions when problem data are described by scenarios instead of point estimates. The authors formulate robust optimization (RO) that explicitly balances solution robustness—remaining close to optimal across all scenarios—and model robustness—remaining almost feasible across all scenarios, and compare it to sensitivity analysis and stochastic linear programming, illustrating the approach with the diet problem and developing RO models for several real‑world applications.

Abstract

Mathematical programming models with noisy, erroneous, or incomplete data are common in operations research applications. Difficulties with such data are typically dealt with reactively—through sensitivity analysis—or proactively—through stochastic programming formulations. In this paper, we characterize the desirable properties of a solution to models, when the problem data are described by a set of scenarios for their value, instead of using point estimates. A solution to an optimization model is defined as: solution robust if it remains “close” to optimal for all scenarios of the input data, and model robust if it remains “almost” feasible for all data scenarios. We then develop a general model formulation, called robust optimization (RO), that explicitly incorporates the conflicting objectives of solution and model robustness. Robust optimization is compared with the traditional approaches of sensitivity analysis and stochastic linear programming. The classical diet problem illustrates the issues. Robust optimization models are then developed for several real-world applications: power capacity expansion; matrix balancing and image reconstruction; air-force airline scheduling; scenario immunization for financial planning; and minimum weight structural design. We also comment on the suitability of parallel and distributed computer architectures for the solution of robust optimization models.