Classical Robust Optimization Research Articles

Distribution-dependent robust linear optimization with applications to inventory control.

This paper tackles linear programming problems with data uncertainty and applies it to an important inventory control problem. Each element of the constraint matrix is subject to uncertainty and is modeled as a random variable with a bounded support. The classical robust optimization approach to this problem yields a solution with guaranteed feasibility. As this approach tends to be too conservative when applications can tolerate a small chance of infeasibility, one would be interested in obtaining a less conservative solution with a certain probabilistic guarantee of feasibility. A robust formulation in the literature produces such a solution, but it does not use any distributional information on the uncertain data. In this work, we show that the use of distributional information leads to an equally robust solution (i.e., under the same probabilistic guarantee of feasibility) but with a better objective value. In particular, by exploiting distributional information, we establish stronger upper bounds on the constraint violation probability of a solution. These bounds enable us to "inject" less conservatism into the formulation, which in turn yields a more cost-effective solution (by 50% or more in some numerical instances). To illustrate the effectiveness of our methodology, we consider a discrete-time stochastic inventory control problem with certain quality of service constraints. Numerical tests demonstrate that the use of distributional information in the robust optimization of the inventory control problem results in 36%-54% cost savings, compared to the case where such information is not used.

Recoverable robust knapsacks: the discrete scenario case

The knapsack problem is one of the basic problems in combinatorial optimization. In real-world applications it is often part of a more complex problem. Examples are machine capacities in production planning or bandwidth restrictions in telecommunication network design. Due to unpredictable future settings or erroneous data, parameters of such a subproblem are subject to uncertainties. In high risk situations a robust approach should be chosen to deal with these uncertainties. Unfortunately, classical robust optimization outputs solutions with little profit by prohibiting any adaption of the solution when the actual realization of the uncertain parameters is known. This ignores the fact that in most settings minor changes to a previously determined solution are possible. To overcome these drawbacks we allow a limited recovery of a previously fixed item set as soon as the data are known by deleting at most k items and adding up to l new items. We consider the complexity status of this recoverable robust knapsack problem and extend the classical concept of cover inequalities to obtain stronger polyhedral descriptions. Finally, we present two extensive computational studies to investigate the influence of parameters k and l to the objective and evaluate the effectiveness of our new class of valid inequalities.

Distribution-Dependent Robust Linear Optimization with Asymmetric Uncertainty and Application to Optimal Control

We consider a linear programming problem in which the constraint matrix is uncertain. Each element of the constraint matrix is modeled as a random variable whose range is asymmetrically bounded around its mean. We construct a formulation that yields a solution with a better objective value, compared to the classical robust optimization approach, while taking the risk that the solution may become infeasible to the original problem. We address the risk by establishing upper bounds on the probability that it violates the constraints of the problem. These bounds exploit full distributional information on the random elements or limited distributional information such as the true means or sample means of the random elements. We explore the application of our methodology to the optimal control of linear uncertain systems with constraints.