Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms

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 O(k(n - k) + n log n) time. This improves upon the result of O(n(2)k) in Bader (2009), and slightly improves upon the result of O(nk + n log n) 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.