Lagrangian Reformulation for Nonconvex Optimization: Tailoring Problems to Specialized Solvers

TL;DR

This paper explores reformulating nonconvex optimization problems, especially with binary variables, using Lagrangian relaxation to leverage advanced solvers and emerging computing technologies.

Contribution

It establishes the equivalence between equality-constrained nonconvex problems and their Lagrangian relaxations, filling a key gap in the literature.

Findings

Characterizes the equivalence between constrained problems and Lagrangian relaxation.

Bridges classical duality results with specific nonconvex problem features.

Provides a comprehensive analysis of problem formulation and solution equivalence.

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.