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In this paper, we present the first outer approximation algorithm formulti-objective mixed-integer linear programming problems with any number ofobjectives. The algorithm also works for certain classes of non-linearprogramming problems. It produces the non-dominated extreme points as well asthe facets of the convex hull of these points. The algorithm relies on anoracle which solves single-objective weighted-sum problems and we show that therequired number of oracle calls is polynomial in the number of facets of theconvex hull of the non-dominated extreme points in the case of multiobjectivemixed-integer programming (MOMILP). Thus, for MOMILP problems for which theweighted-sum problem is solvable in polynomial time, the facets can be computedwith incremental-polynomial delay. From a practical perspective, the algorithmstarts from a valid lower bound set for the non-dominated extreme points anditeratively improves it. Therefore it can be used in multi-objectivebranch-and-bound algorithms and still provide a valid bound set at any stage,even if interrupted before converging. Moreover, the oracle produces Paretooptimal solutions, which makes the algorithm also attractive from the primalside in a multi-objective branch-and-bound context. Finally, the oracle canalso be called with any relaxation of the primal problem, and the obtainedpoints and facets still provide a valid lower bound set. A computational studyon a set of benchmark instances from the literature and new non-linearmulti-objective instances is provided.

An outer approximation algorithm for multi-objective mixed-integer linear and non-linear programming