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 time. This improves upon the result of in Bader ( 2009 ), and slightly improves upon the result of 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 .

Given a nondominated point set of size and a suitable reference point , the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of . In contrast, the best upper bound available for is . Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of using a greedy strategy. In this article, greedy -time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem.

In this article, a local search approach is proposed for three variants of the bi-objective binary knapsack problem, with the aim of maximizing the total profit and minimizing the total weight. First, an experimental study on a given structural property of connectedness of the efficient set is conducted. Based on this property, a local search algorithm is proposed and its performance is compared to exact algorithms in terms of runtime and quality metrics. The experimental results indicate that this simple local search algorithm is able to find a representative set of optimal solutions in most of the cases, and in much less time than exact algorithms.