An outer approximation algorithm for multiobjective mixed-integer linear programming.

In this work, we present the first outer approximation algorithm for multiobjective mixed-integer linear programming (MOMILP). The algorithm produces the non-dominated extreme points for MOMILP, and the facets of the convex hull of these points. Our algorithm uses an oracle which consists of solving single-objective weighted-sum problems. We show that analogous to existing outer approximation algorithms for multiobjective linear programming, our algorithm needs a number of oracle calls, which is polynomial in the number of facets of the convex hull of the non-dominated extreme points. As a consequence, for problems for which the weighted-sum problem is solvable in polynomial time, the facets can be computed in incremental polynomial time. From a practical perspective, due to its outer approximation nature, the algorithm provides valid lower bound sets for use in multiobjective branch-and-bound algorithms at any stage of its execution and thus allows for early termination of such a bound computation. Moreover, the oracle produces Pareto optimal solutions, which makes the algorithm also attractive from the primal side. Finally, the oracle can also be called with any relaxation of the original MOMILP, and the obtained points and facets still provide a valid lower bound set. A computational study is done to assess the efficiency of our algorithm.