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.

A novel parallel decomposition algorithm is developed for large, multistage stochastic optimization problems. The method decomposes the problem into subproblems that correspond to scenarios. The subproblems are modified by separable quadratic terms to coordinate the scenario solutions. Convergence of the coordination procedure is proven for linear programs. Subproblems are solved using a nonlinear interior point algorithm. The approach adjusts the degree of decomposition to fit the available hardware environment. Initial testing on a distributed network of workstations shows that an optimal number of computers depends upon the work per subproblem and its relation to the communication capacities. The algorithm has promise for solving stochastic programs that lie outside current capabilities.

Several financial planning problems are posed as dynamic generalized network models with stochastic parameters. Examples include: asset allocation for portfolio selection, international cash management, and programmed-trading arbitrage. Despite the large size of the resulting stochastic programs, the network structure can be exploited within the solution strategy giving rise to efficient implementations. Empirical results are presented indicating the benefits of the stochastic network approach for the asset allocation case.financial planning, stochastic programming, generalized networks, scenario analysis

The problem of designing contaminated groundwater remediation systems using hydraulic control is addressed. Two nonlinear optimization formulations are proposed which model the design process for the location and pump rates of injection and extraction wells in an aquifer cleanup system. The formulations are designed to find a pumping system which (1) removes the most contaminant over a fixed time period and (2) reduces contaminant concentration to specified levels by the end of a fixed time period at least cost. The formulations employ a two‐dimensional Galerkin finite element simulation model of steady state groundwater flow and transient convective‐dispersive transport. To make the optimization problems computationally tractable sensitivity theory is used to derive a general relationship for computing the derivatives of an arbitrary function of the simulation outputs with respect to model inputs. This relationship is then applied to the convective‐dispersive transport equation.

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