title: Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms.
abstract: The hypervolume subset selection problem consists of finding a subset, with a given cardinality k, of a set of nondominated points that maximizes the hypervolume indicator. This problem arises in selection procedures of evolutionary algorithms for multiobjective optimization, for which practically efficient algorithms are required. In this article, two new formulations are provided for the two-dimensional variant of this problem. The first is a (linear) integer programming formulation that can be solved by solving its linear programming relaxation. The second formulation is a k-link shortest path formulation on a special digraph with the Monge property that can be solved by dynamic programming in [Formula: see text] time. This improves upon the result of [Formula: see text] in Bader ( 2009 ), and slightly improves upon the result of [Formula: see text] in Bringmann et al. ( 2014b ), which was developed independently from this work using different techniques. Numerical results are shown for several values of n and k.
authors.affiliation: Mathematical Institute, University of Kaiserslautern, Germany.
authors.forename: Tobias
authors.initials: T
authors.lastname: Kuhn
authors.affiliation: CISUC, Department of Informatics Engineering, University of Coimbra, Portugal.
authors.forename: Carlos M
authors.initials: CM
authors.lastname: Fonseca
authors.affiliation: CISUC, Department of Informatics Engineering, University of Coimbra, Portugal.
authors.forename: Luís
authors.initials: L
authors.lastname: Paquete
authors.affiliation: Mathematical Institute, University of Koblenz-Landau, Campus Koblenz, Germany.
authors.forename: Stefan
authors.initials: S
authors.lastname: Ruzika
authors.affiliation: CISUC, Department of Informatics Engineering, University of Coimbra, Portugal.
authors.forename: Miguel M
authors.initials: MM
authors.lastname: Duarte
authors.affiliation: CEG-IST, Instituto Superior Técnico, Universidade de Lisboa, Portugal.
authors.forename: José Rui
authors.initials: JR
authors.lastname: Figueira
mesh_terms: D000465:Algorithms; D008962:Models, Theoretical
keywords: Multiobjective optimization; hypervolume; k-link shortest path; subset selection