Robust Optimization of Large-Scale Systems

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.

Risk management policies for dynamic capacity control

“…Mulvey et al [18] propose a different approach and consider robustness of solutions in a set of scenarios. They introduce a penalty function to the objective function to achieve a tradeoff between optimality and feasibility.…”

Models and algorithms for network design problems

“…Note that this problem can also be seen as a robust knapsack problem with linear penalty (Mulvey et al, 1995). With C uniformly distributed, (5.5) becomes: …”

Determining production level under uncertainty using fuzzy simulation and bootstrap technique, a case study

“…Mula et al (2006) have provided a review of literature in production planning under uncertainty. Stochastic programming (Dantzig 1955;Kall and Wallace 1994;Birge and Louveaux 1997;Kall and Mayer 2005) and robust optimization (Mulvey et al 1995) has seen several successful applications in production planning. In Escudero et al (1993), a multi-stage stochastic programming approach was used for addressing a MPMP production planning model with random demand.…”