Skip to main content
SpringerLink
Log in

Automated Aggregation and Omission of Objectives for Tackling Many-Objective Problems

  • Conference paper
  • First Online:
New Developments in Multiple Objective and Goal Programming

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 638))

Abstract

Many-objective problems pose various challenges in terms of decision making and search. One approach to tackle the resulting problems is the automatic reduction of the number of objectives such that the information loss is minimized. While in a previous work we have investigated the issue of omitting objectives, we here address the generalized problem of aggregating objectives using weighted sums. To this end, heuristics are presented that iteratively remove two objectives and replace them by a new objective representing an optimally weighted combination of them. As shown in the paper, the new reduction method can substantially reduce the information loss and thereby can be highly useful when analyzing trade-off sets after optimization as well as during search to reduce the computation overhead related to hypervolume-based fitness assignment.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This is the case, e.g., in hypervolume-based evolutionary algorithms where the computation time per generation is exponential in the number of objectives (Brockhoff and :̧def :̧def Zitzler 2007a; :̧def :̧def Bringmann and Friedrich 2008). Also if the objective values have to be computed by time-consuming simulations, a lower number of objectives will speed up the evaluation time.

  2. 2.

    For simplicity, we use the term objective and the corresponding weight vector interchangeably throughout the paper.

  3. 3.

    More precisely, to 1 and 2 objectives for the 4-objective instances, to 1 and 3 for the 6-objective instances, and to 2 and 4 for the 8-objective instances.

  4. 4.

    The number of decision variables has been set to 250.

  5. 5.

    For all WFG problems, the number of decision variables has been also fixed to 250 and the number of position variables has been chosen to 168 and the number of distance variables to 82 such that it can be kept constant over all numbers of objectives.

  6. 6.

    The δ-errors have been normalized to the objective values of each instance such that the difference between the highest and lowest objective value equals 1 for every objective.

  7. 7.

    The \(\#\mathcal{P}\)-hardness proof of :̧def :̧def Bringmann and Friedrich (2008) implies that no exact polynomial algorithm for the hypervolume indicator exists unless \(\mathcal{P} = \mathcal{N}\mathcal{P}\).

  8. 8.

    The hypervolume indicator or \(\mathcal{S}\)-measure of a solution set A ⊂ X is informally defined as the space that is dominated by the solutions in A which itself is dominating a reference set. Here, we use only a single reference point and refer to Beume et al. (2007) for an exact definition. The hypervolume indicator has always to be maximized.

References

  • Beume N, Naujoks B, Emmerich M (2007) SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur J Oper Res 181:1653–1669

    Article  Google Scholar 

  • Bleuler S, Laumanns M, Thiele L, Zitzler E (2003) PISA – a platform and programming language independent interface for search algorithms. In: Fonseca CM, et al (eds) Conference on evolutionary multi-criterion optimization (EMO 2003). LNCS, vol 2632. Springer, Berlin, pp 494–508

    Chapter  Google Scholar 

  • Bringmann K, Friedrich T (2008) Approximating the volume of unions and intersections of high-dimensional geometric objects. In: Hong SH, Nagamochi H, Fukunaga T (eds) International symposium on algorithms and computation (ISAAC 2008). LNCS, vol 5369. Springer, Berlin, pp 436–447

    Google Scholar 

  • Brockhoff D, Zitzler E (2007a) Improving hypervolume-based multiobjective evolutionary algorithms by using objective reduction methods. In: Congress on evolutionary computation (CEC 2007). IEEE Press, pp 2086–2093

    Google Scholar 

  • Brockhoff D, Zitzler E (2007b) Offline and online objective reduction in evolutionary multiobjective optimization based on objective conflicts. TIK Report 269, Computer Engineering and Networks Laboratory (TIK), ETH Zurich

    Google Scholar 

  • Brockhoff D, Zitzler E (2009) Objective reduction in evolutionary multiobjective optimization: theory and applications. Evol Comput 17(2):135–166

    Article  Google Scholar 

  • Brockhoff D, Friedrich T, Hebbinghaus N, Klein C, Neumann F, Zitzler E (2007) Do additional objectives make a problem harder? In: Thierens D, et al (eds) Genetic and evolutionary computation conference (GECCO 2007). ACM Press, New York, NY, USA, pp 765–772

    Chapter  Google Scholar 

  • Conover WJ (1999) Practical nonparametric statistics, 3rd edn. Wiley, New York

    Google Scholar 

  • Deb K, Saxena D (2006) Searching for Pareto-optimal solutions through dimensionality reduction for certain large-dimensional multi-objective optimization problems. In: Congress on evolutionary computation (CEC 2006). IEEE Press, pp 3352–3360

    Google Scholar 

  • Deb K, Thiele L, Laumanns M, Zitzler E (2005) Scalable test problems for evolutionary multi-objective optimization. In: Abraham A, Jain R, Goldberg R (eds) Evolutionary multiobjective optimization: theoretical advances and applications, chap 6. Springer, Berlin, pp 105–145

    Chapter  Google Scholar 

  • Fleming PJ, Purshouse RC, Lygoe RJ (2005) Many-objective optimization: an engineering design perspective. In: Coello Coello CA, et al (eds) Conference on evolutionary multi-criterion optimization (EMO 2005). LNCS, vol 3410. Springer, Berlin, pp 14–32

    Chapter  Google Scholar 

  • Huband S, Hingston P, Barone L, While L (2006) A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans Evol Comput 10(5):477–506

    Article  Google Scholar 

  • Hughes EJ (2007) Radar waveform optimization as a many-objective application benchmark. In: Conference on evolutionary multi-criterion optimization (EMO 2007). LNCS, vol 4403. Springer, Berlin, pp 700–714

    Google Scholar 

  • Künzli S, Bleuler S, Thiele L, Zitzler E (2004) A computer engineering benchmark application for multiobjective optimizers. In: Coello Coello CA, Lamont G (eds) Applications of multi-objective evolutionary algorithms. World Scientific, Singapore, pp 269–294

    Google Scholar 

  • Laumanns M, Thiele L, Deb K, Zitzler E (2002) Combining convergence and diversity in evolutionary multiobjective optimization. Evol Comput 10(3):263–282

    Article  Google Scholar 

  • Laumanns M, Thiele L, Zitzler E (2004) Running time analysis of evolutionary algorithms on a simplified multiobjective Knapsack problem. Nat Comput 3(1):37–51

    Article  Google Scholar 

  • López Jaimes A, Coello Coello CA, Chakraborty D (2008) Objective reduction using a feature selection technique. In: Conference on genetic and evolutionary computation (GECCO 2008). ACM, New York, USA, pp 673–680, doi http://doi.acm.org/10.1145/1389095.1389228

    Google Scholar 

  • Purshouse RC, Fleming PJ (2003) Conflict, harmony, and independence: relationships in evolutionary multi-criterion optimisation. In: Conference on evolutionary multi-criterion optimization (EMO 2003). LNCS, vol 2632. Springer, Berlin, pp 16–30

    Google Scholar 

  • Saxena DK, Deb K (2007) Non-linear dimensionality reduction procedures for certain large-dimensional multi-objective optimization problems: employing correntropy and a novel maximum variance unfolding. In: Conference on evolutionary multi-criterion optimization (EMO 2007). LNCS, vol 4403. Springer, Berlin, pp 772–787, doi 10.1007/ 978-3-540-70928-2_58

    Google Scholar 

  • Sülflow A, Drechsler N, Drechsler R (2007) Robust multi-objective optimization in high dimensional spaces. In: Conference on evolutionary multi-criterion optimization (EMO 2007). LNCS, vol 4403. Springer, Berlin, pp 715–726, doi 10.1007/ 978-3-540-70928-2_54

    Google Scholar 

  • Zitzler E, Künzli S (2004) Indicator-based selection in multiobjective search. In: Yao X, et al (eds) Conference on parallel problem solving from nature (PPSN VIII). LNCS, vol 3242. Springer, Berlin, pp 832–842

    Chapter  Google Scholar 

  • Zitzler E, Thiele L (1998) Multiobjective optimization using evolutionary algorithms – a comparative case study. In: Conference on parallel problem solving from nature (PPSN V), Amsterdam, pp 292–301

    Google Scholar 

  • Zitzler E, Brockhoff D, Thiele L (2007) The hypervolume indicator revisited: on the design of Pareto-compliant indicators via weighted integration. In: Obayashi S, et al (eds) Conference on evolutionary multi-criterion optimization (EMO 2007). LNCS, vol 4403. Springer, Berlin, pp 862–876

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Johannes Bader for the support with the illustrations. Dimo Brockhoff has been supported by the Swiss National Science Foundation (SNF) under grant 112079.

Author information

Authors and Affiliations

  1. Computer Engineering and Networks Laboratory, ETH Zurich, 8092, Zurich, Switzerland

    Dimo Brockhoff

Authors
  1. Dimo Brockhoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eckart Zitzler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dimo Brockhoff .

Editor information

Editors and Affiliations

  1. University of Portsmouth, Lion Terrace, Portsmouth, PO1 3HE, United Kingdom

    Dylan Jones

  2. College of Business & Administration, Kuwait University, Safat, Kuwait

    Mehrdad Tamiz

  3. Dept. Mathematics, University of Portsmouth, Lion Terrace, Portsmouth, PO1 3HE, United Kingdom

    Jana Ries

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Brockhoff, D., Zitzler, E. (2010). Automated Aggregation and Omission of Objectives for Tackling Many-Objective Problems. In: Jones, D., Tamiz, M., Ries, J. (eds) New Developments in Multiple Objective and Goal Programming. Lecture Notes in Economics and Mathematical Systems, vol 638. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10354-4_6

Download citation

Publish with us

Policies and ethics

Search

Navigation