Lagrangian Reformulation for Nonconvex Optimization: Tailoring Problems to Specialized Solvers
Rodolfo A. Quintero, Juan C. Vera, Luis F. Zuluaga
TL;DR
This work develops a constructive theory linking equality-constrained nonconvex optimization to Lagrangian relaxations, proving strong duality and dual-attainment conditions that yield exact reformulations. It provides practical pathways to tailor problems for specialized solvers, including QUBO and mixed-integer reformulations suitable for Ising devices and quantum algorithms. The results extend beyond linear constraints to nonlinear equalities, and include convergence guarantees for penalized formulations and rounding schemes to recover integer solutions. The framework thus enables problem reformulations that harness emerging hardware and solver paradigms with broad applicability across combinatorial and mixed-integer optimization tasks.
Abstract
In recent years, there has been a surge of interest in studying different ways to reformulate nonconvex optimization problems, especially those that involve binary variables. This interest surge is due to advancements in computing technologies, such as quantum and Ising devices, as well as improvements in quantum and classical optimization solvers that take advantage of particular formulations of nonconvex problems to tackle their solutions. Our research characterizes the equivalence between equality-constrained nonconvex optimization problems and their Lagrangian relaxation, enabling the aforementioned new technologies to solve these problems. In addition to filling a crucial gap in the literature, our results are readily applicable to many important situations in practice. To obtain these results, we bridge between specific optimization problem characteristics and broader, classical results on Lagrangian duality for general nonconvex problems. Further, our approach takes a comprehensive approach to the question of equivalence between problem formulations. We consider this question not only from the perspective of the problem's objective but also from the viewpoint of its solution. This perspective, often overlooked in existing literature, is particularly relevant for problems featuring continuous and binary variables.