On the Relationship Between the Value Function and the Efficient Frontier of a Mixed Integer Linear Optimization Problem

TL;DR

This paper explores the mathematical relationship between the efficient frontier of multiobjective MILPs and the restricted value function of related single-objective MILPs, proposing a new algorithm for constructing the EF.

Contribution

It establishes a theoretical link between the EF and the RVF in MILPs and introduces a generalized cutting-plane algorithm for EF construction based on VF methods.

Findings

The EF corresponds to the boundary of the RVF's epigraph.

Methods for constructing the VF and EF are interchangeable.

The proposed algorithm converges finitely and has performance guarantees.

Abstract

In this study, we investigate the connection between the efficient frontier (EF) of a general multiobjective mixed integer linear optimization problem (MILP) and the so-called restricted value function (RVF) of a closely related single-objective MILP. In the first part of the paper, we detail the mathematical structure of the RVF, including characterizing the set of points at which it is differentiable, the gradients at such points, and the subdifferential at all nondifferentiable points. We then show that the EF of the multiobjective MILP is comprised of points on the boundary of the epigraph of the RVF and that any description of the EF suffices to describe the RVF and vice versa. Because of the close relationship of the RVF to the EF, we observe that methods for constructing the so-called value function (VF) of an MILP and methods for constructing the EF of a multiobjective optimization problem are effectively interchangeable. Exploiting this observation, we propose a generalized cutting-plane algorithm for constructing the EF of a multiobjective MILP that arises from an existing algorithm for constructing the classical MILP VF. The algorithm identifies the set of all integer parts of solutions on the EF. We prove that the algorithm converges finitely under a standard boundedness assumption and comes with a performance guarantee if terminated early.