A note on the ϵ -indicator subset selection

Abstract

The -indicator subset selection selects a subset of a nondominated point set that is as close as possible to a reference point set with respect to the -indicator. This selection procedure is used by population-based heuristic approaches for multiobjective optimization problems. Given that this procedure is called very often during the run of the heuristic approach, efficient ways of computing the optimal subset are strongly required. In this note, we give a correctness proof of the -indicator subset selection algorithm proposed by Ponte et al. [1] for the bidimensional case as well as several algorithmic improvements in terms of time complexity. Extensions to larger dimension are also discussed.

Two-dimensional subset selection for hypervolume and epsilon-indicator

“…This answers an open problem of Vaz et al [19], of whether there is a subquadratic time algorithm for EPSSSP (although they might have intended a deterministic algorithm, while we present a randomized one). Again, this improves the state of the art by roughly a factor of n (at least if n = m), which is confirmed by preliminary experiments (cf.…”

“…As has been observed in [18,19], ε * is among the values I eps (p, r) with p ∈ P , r ∈ R, since at least one pair (p, r) prevents us from further decreasing ε * . We can split these values into sorted sequences as follows.…”

“…For the ε-indicator we set P = R so that n = m. We implemented the O(n 2 log n) algorithm (Algorithm 3), which has the same asymptotic runtime as [18,19], and compared it with both algorithms we described (Algorithm 6 and Algorithm 7).…”

“…EPSSSP was first studied by Ponte, Paquete, and Figueira [18], who gave an O(nm log(nm)) time algorithm for dimension d = 2. Recently, Vaz, Paquete, and Ponte [19] slightly improved this algorithm by lowering the log-factor, however, the runtime remains Ω(nm). Again, in higher dimensions, no algorithm is known that is faster than enumerating all subsets of P of size k.…”

Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms

“…In each iteration a pool of solutions is determined and a subset of these solutions with fixed cardinality is kept for further iterations. Hence, fast subset selection procedures can enhance the running time of such heuristic approaches (see [10,11] for the case of the -indicator).…”