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Relaxations and Duality for Multiobjective Integer Programming
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
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