Technical Note—Dual Approach for Two-Stage Robust Nonlinear Optimization

In “Dual Approach for Two-Stage Robust Nonlinear Optimization,” de Ruiter, Zhen, and den Hertog study adjustable robust minimization problems where the objective or constraints depend in a convex way on the adjustable variables. They reformulate the original adjustable robust nonlinear problem with a polyhedral uncertainty set into an equivalent adjustable robust linear problem, for which all existing approaches for adjustable robust linear problems can be used. The reformulation is obtained by first dualizing over the adjustable variables and then over the uncertain parameters. The polyhedral structure of the uncertainty set then appears in the linear constraints of the dualized problem, and the nonlinear functions of the adjustable variables in the original problem appear in the uncertainty set of the dualized problem. The authors show how to recover linear decision rules to the original primal problem and how to generate bounds on its optimal objective value. Adjustable robust minimization problems where the objective or constraints depend in a convex way on the adjustable variables are generally difficult to solve. In this paper, we reformulate the original adjustable robust nonlinear problem with a polyhedral uncertainty set into an equivalent adjustable robust linear problem, for which all existing approaches for adjustable robust linear problems can be used. The reformulation is obtained by first dualizing over the adjustable variables and then over the uncertain parameters. The polyhedral structure of the uncertainty set then appears in the linear constraints of the dualized problem, and the nonlinear functions of the adjustable variables in the original problem appear in the uncertainty set of the dualized problem. We show how to recover linear decision rules to the original primal problem and how to generate bounds on its optimal objective value. Funding: The research of F. J. C. T. de Ruiter is partially supported by the Netherlands Organisation for Scientific Research (NWO) [Talent Grant 406-14-067]. J. Zhen is partially supported by the NWO [Grant 613.001.208]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.2289 .