On a practical notion of Geoffrion proper optimality in multicriteria optimization

Geoffrion proper optimality is a widely used optimality notion in multicriteria optimization that prevents exact solutions having unbounded trade-offs. As algorithms for multicriteria optimization usually give only approximate solutions, we analyze the notion of approximate Geoffrion proper optimality. We show that in the limit, approximate Geoffrion proper optimality may converge to solutions having unbounded trade-offs. Therefore, we introduce a restricted notion of approximate Geoffrion proper optimality and prove that this restricted notion alleviates the problem of solutions having unbounded trade-offs. Furthermore, using a characterization based on the infeasibility of a system of inequalities, we investigate the convergence properties of different approximate optimality notions in multicriteria optimization. These convergence properties are important for algorithmic reasons. The restricted notion of approximate Geoffrion proper optimality seems to be the only approximate optimality notion that shows favourable convergence properties. This notion bounds the trade-offs globally and can be used in multicriteria decision-making algorithms as well. Due to these, it seems to be a practical optimality notion.