Robust and Reliable Solutions in Bilevel Optimization Problems Under Uncertainties

Abstract Bilevel optimization problems have received a growing attention in the recent past due to their relevance in practice. A number of studies on bilevel applications and solution methodologies are available for deterministic setup, but studies on uncertainties in bilevel optimization are rare. In this paper, we suggest methodologies for handling uncertainty in both lower and upper level decision variables that may occur from different practicalities. For the first time, we discuss and demonstrate the effect of uncertainties in each level along with the definition of a robustness and reliability in the context of bilevel optimization. To begin with, the definitions of robustness and reliability in the presence of uncertain upper and lower level variables are introduced. The issues and complexities introduced due to such uncertainties are then studied through a number of test cases. Finally, two real world bilevel test problems involving uncertainties in decision variables are solved. The first problem is an electrical circuit design problem where the objective is to maximize the gain at upper level with a stability requirement at the lower level. The second problem is a constrained navy ship design problem involving physical design parameters at the upper level and operating parameters at the lower level.