Relaxations and Duality for Multiobjective Integer Programming

TL;DR

This paper explores relaxations and duality in multiobjective integer programming, introducing new dual formulations and demonstrating their effectiveness in providing tighter bounds and insights into supported and unsupported solutions.

Contribution

It introduces a Lagrangian dual and a set-valued superadditive dual for MOIPs, extending duality theory and improving bound quality over existing relaxations.

Findings

Lagrangian relaxation can provide tighter bounds than convex hull relaxation at unsupported solutions.

The Lagrangian dual satisfies weak duality and is strong at supported solutions under certain conditions.

A set-valued superadditive dual is proposed, being strong at supported solutions and weak in general.

Abstract

Multiobjective integer programs (MOIPs) simultaneously optimize multiple objective functions over a set of linear constraints and integer variables. In this paper, we present continuous, convex hull and Lagrangian relaxations for MOIPs and examine the relationship among them. The convex hull relaxation is tight at supported solutions, i.e., those that can be derived via a weighted-sum scalarization of the MOIP. At unsupported solutions, the convex hull relaxation is not tight and a Lagrangian relaxation may provide a tighter bound. Using the Lagrangian relaxation, we define a Lagrangian dual of an MOIP that satisfies weak duality and is strong at supported solutions under certain conditions on the primal feasible region. We include a numerical experiment to illustrate that bound sets obtained via Lagrangian duality may yield tighter bounds than those from a convex hull relaxation. Subsequently, we generalize the integer programming value function to MOIPs and use its properties to motivate a set-valued superadditive dual that is strong at supported solutions. We also define a simpler vector-valued superadditive dual that exhibits weak duality but is strongly dual if and only if the primal has a unique nondominated point.