{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T02:14:42Z","timestamp":1774836882154,"version":"3.50.1"},"reference-count":23,"publisher":"MIT Press - Journals","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Evolutionary Computation"],"published-print":{"date-parts":[[2015,3]]},"abstract":"<jats:p> Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multi-objective problems as the population of such an algorithm can be used to represent the trade-offs with respect to the given objective functions. In this paper, we contribute to the theoretical understanding of evolutionary algorithms for multi-objective problems. We consider indicator-based algorithms whose goal is to maximize the hypervolume for a given problem by distributing [Formula: see text] points on the Pareto front. To gain new theoretical insights into the behavior of hypervolume-based algorithms, we compare their optimization goal to the goal of achieving an optimal multiplicative approximation ratio. Our studies are carried out for different Pareto front shapes of bi-objective problems. For the class of linear fronts and a class of convex fronts, we prove that maximizing the hypervolume gives the best possible approximation ratio when assuming that the extreme points have to be included in both distributions of the points on the Pareto front. Furthermore, we investigate the choice of the reference point on the approximation behavior of hypervolume-based approaches and examine Pareto fronts of different shapes by numerical calculations. <\/jats:p>","DOI":"10.1162\/evco_a_00126","type":"journal-article","created":{"date-parts":[[2014,3,21]],"date-time":"2014-03-21T14:20:46Z","timestamp":1395411646000},"page":"131-159","source":"Crossref","is-referenced-by-count":14,"title":["Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point"],"prefix":"10.1162","volume":"23","author":[{"given":"Tobias","family":"Friedrich","sequence":"first","affiliation":[{"name":"Lehrstuhl Theoretische Informatik I, Fakult\u00e4t f\u00fcr Mathematik und Informatik, Friedrich-Schiller-Universit\u00e4t Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany"}]},{"given":"Frank","family":"Neumann","sequence":"additional","affiliation":[{"name":"Optimisation and Logistics, School of Computer Science, University of Adelaide, Adelaide, SA 5005, Australia"}]},{"given":"Christian","family":"Thyssen","sequence":"additional","affiliation":[{"name":"Lehrstuhl 2, Fakult\u00e4t f\u00fcr Informatik, Technische Universit\u00e4t Dortmund, 44221 Dortmund, Germany"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1145\/1527125.1527138"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1201\/9781420050387.ptb"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1162\/EVCO_a_00009"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2006.08.008"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2011.12.001"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2010.03.004"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2010.09.026"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1016\/j.artint.2012.09.005"},{"key":"B10","unstructured":"Brockhoff, D. (2010). Optimal -distributions for the hypervolume indicator for problems with linear bi-objective fronts: Exact and exhaustive results. In Proceeding of the 8th International Conference on Simulated Evolution and Learning, SEAL \u201910, Lecture notes in computer science, Vol. 6457 (pp. 24\u201334). 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