On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

TL;DR

This paper explores nonconvex surrogate relaxations for mixed-integer nonlinear programs, demonstrating their potential to produce tighter bounds and introducing a novel algorithm for optimal aggregation sets.

Contribution

It revisits surrogate relaxations in MINLPs, generalizes them for multiple aggregations, and develops the first algorithm to compute optimal aggregation sets with practical enhancements.

Findings

Nonconvex relaxations can yield tighter bounds in MINLPs.

The proposed algorithm effectively finds the best set of aggregations.

Computational experiments show improved dual bounds using the new method.

Abstract

The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm's ability to generate strong dual bounds through extensive computational experiments.