A coverage-based Box-Algorithm to compute a representation for optimization problems with three objective functions

Abstract

Abstract. A new algorithm for optimization problems with three objective functions is presented which computes a representation for the set of nondominated points. This representation is guaranteed to have a desired coverage error and a bound on the number of iterations needed by the algorithm to meet this coverage error is derived. Since the representation does not necessarily contain nondominated points only, ideas to calculate bounds for the representation error are given. Moreover, the incorporation of domination during the algorithm and other quality measures are discussed.

Equidistant representations: Connecting coverage and uniformity in discrete biobjective optimization

“…This was done in order to create tighter boxes in each step for the purpose of developing an efficient method for generating the entire nondominated set instead of a representation. The box-method was also recently extended to the problem of generating representations for problems with three objectives by Kuhn and Ruzika (2016). Sylva and Crema (2007) and Masin and Bukchin (2008) independently propose essentially identical methods for MIPs with two (or more) objectives, where during each iteration the point that maximizes the ∞-norm distance to the current dominated region (defined by the current representation) is generated and added to the representation.…”

“…Faulkenberg and Wiecek (2010) present a comprehensive survey of more than twenty different ones. However, the approach originally proposed by Sayın (2000) has become a standard way of evaluating representations (Eusébio et al, 2014;Vaz et al, 2015;Kuhn and Ruzika, 2016;Shao and Ehrgott, 2016). We adopt this approach in this paper as well.…”

Supported nondominated points as a representation of the nondominated set: An empirical analysis

“…On a separate vein, the idea of finding representations of the nondominated set that conform to some quality guarantees was proposed as an alternative to computing the entire nondominated set in Sayın (2000). Promising algorithms were presented for the bicriteria case in Sayın and Kouvelis (2005), Kouvelis and Sayın (2006), Hamacher et al (2007) and for the three criteria case in Kuhn and Ruzika (2017). Further generalization of these ideas to higher dimensions has proven to be computationally challenging when rigorous quality guarantees are A question that has not been adequately answered so far is whether, despite its low cardinality, the set of supported nondominated points constitute a fine representation of the entire nondominated set.…”