In this paper, 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. We demonstrate that the EF of the multiobjective MILP is
comprised of points on the boundary of the epigraph of the RVF so that any
description of the EF suffices to describe the RVF and vice versa. In the first
part of the paper, we describe 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.
Because of the close relationship of the RVF to the EF, we observe that methods
for constructing so-called value functions and methods for constructing the EF
of a multiobjective optimization problem, each of which have been developed in
separate communities, are effectively interchangeable. By exploiting this
relationship, we propose a generalized cutting plane algorithm for constructing
the EF of a multiobjective MILP based on a generalization of an existing
algorithm for constructing the classical value function. We prove that the
algorithm is finite under a standard boundedness assumption and comes with a
performance guarantee if terminated early