Solving two-level optimization problems with applications to robust design and energy markets

TL;DR: This dissertation provides a method to decompose and solve this two-level structure using a modified Benders decomposition, applicable to robust optimization problems with quasiconvex constraints and provides approximate solutions to problems with nonlinear constraints.

Abstract: : This dissertation provides efficient techniques to solve two-level optimization problems. Three specific types of problems are considered. The first problem is robust optimization, which has direct applications to engineering design. Traditionally robust optimization problems have been solved using an inner-outer structure, which can be computationally expensive. This dissertation provides a method to decompose and solve this two-level structure using a modified Benders decomposition. This gradient-based technique is applicable to robust optimization problems with quasiconvex constraints and provides approximate solutions to problems with nonlinear constraints. The second types of two-level problems considered are mathematical and equilibrium programs with equilibrium constraints. Their two-level structure is simplified using Schur's decomposition and reformulation schemes for absolute value functions. The resulting formulations are applicable to game theory problems in operations research and economics. The third type of two-level problem studied is discretely-constrained mixed linear complementarity problems. These are first formulated into a two-level mathematical program with equilibrium constraints and then solved using the aforementioned technique for mathematical and equilibrium programs with equilibrium constraints. The techniques for all three problems help simplify the two-level structure into one level, which helps gain numerical and application insights. The computational effort for solving these problems is greatly reduced using the techniques in this dissertation. Finally, a host of numerical examples are presented to verify the approaches. Diverse applications to economics, operations research, and engineering design motivate the relevance of the novel methods developed in this dissertation.