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Towards Multi-objective Mixed Integer Evolution Strategies Koen van der Blom , Kaifeng Yang, Thomas Bck & Michael Emmerich 21-09-2018 Discover the world at Leiden University Motivation Mixed Integer Evolution Strategy [Li et al 2013]

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Towards Multi-objective Mixed Integer Evolution Strategies

Koen van der Blom, Kaifeng Yang, Thomas Bäck & Michael Emmerich 21-09-2018

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Motivation

  • Mixed Integer Evolution Strategy [Li et al 2013]
  • Real, integer, categorical
  • Single objective
  • Existing multi-objective techniques
  • Weight space decomposition (MILP) [Przybylski et al 2010]
  • Enhanced Directed Search (EDS) [Laredo 2015]
  • Zigzag [Wang 2013, Wang 2015]
  • No distinction between integer and categorical!
  • Extend MIES for multiple objectives

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Evolution Strategies

  • Mimic evolution for optimisation
  • Parents generate offspring (recombination)
  • Offspring add additional variation (mutation)
  • Optimal mutation strength?
  • Changes over time…
  • Step size adaptation
  • Same evolutionary mechanisms!

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(𝝂 + 𝝁) Evolution Strategy [Schwefel 1981]

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  • 𝜈 parents generate 𝜇 offspring
  • Update decision variables
  • Update mutation probabilities
  • Select the best from 𝜈 ∪ 𝜇
  • Repeat
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Mixed Integer Evolution Strategy

[Li et al 2013]

5 Continuous Variables (Normal Distribution) Integer Variables (Geometrical Distribution) Categorical Variables

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Mutation operators

  • Properties
  • Scalability (scale step size)
  • Asymmetry (maximal entropy, avoid bias)
  • Infinite support (every solution is reachable)
  • Example
  • Mutation of integer variables

[Rudolph 1994]

  • Difference of two

Geometric distributions

6 Image from [Li et al 2013]

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Multi-objective optimisation

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Train Routing

  • Time
  • Price

Min Min

TICKET SPEED CHANGE S PRICE OPT1 3:00 H 2 60 EUR OPT2 3:00 H 1 65 EUR OPT3 3:30 H 3 44 EUR OPT4 4:30 H 2 41 EUR OPT5 15:30 4 35 EUR OPT6 15:34 4 32 EUR

Price Time Pareto front

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SMS-EMOA [Emmerich et al 2005]

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  • Optimise hypervolume indicator
  • Rank solutions
  • Non-dominated sorting
  • Hypervolume contribution
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Hypervolume indicator

  • Measure the dominated region

9 Image from [Emmerich+Deutz 2018]

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𝑻-metric (hypervolume) selection

  • Hypervolume contribution

10 Image from [Emmerich+Deutz 2018]

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Non-dominated sorting

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Price Time Pareto front 2nd non-dominated front

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Multi-Objective MIES

  • Canonical MIES operators
  • 𝑇-metric selection
  • Non-dominated sorting
  • (𝜈 + 1) strategy
  • Always select the 𝜈 best
  • (HV never decreases)

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Alternative MO-MIES algorithms

  • Mutation only
  • Best results without recombination [Wessing et al 2017]
  • Different optimal step size for different directions
  • Mutation tournament
  • Greater selection pressure

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Price Time Pareto front

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Scalable Test Problems

  • Multi-sphere
  • Multi-barrier
  • Optical filter
  • Layers (on/off)
  • Thickness per layer

14 Image from [Li et al 2013]

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Experimental setup

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  • 10,000 evaluations
  • 25 repetitions
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Optical filter convergence

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Barrier convergence

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Step size adaptation (multisphere)

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Step size adaptation – Categorical

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Future work

  • Improve categorical step size adaptation
  • Investigate recombination behaviour
  • Why does it work?
  • When will it not work?
  • Introduce multi-objective recombination?
  • Investigate integer step size adaptation
  • Can we prevent regressive behaviour?

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Summary

  • Goal:
  • Extend the MIES algorithm for the multi-objective case
  • Plan:
  • Evaluate MIES + SMS-EMOA (= MOMIES)
  • Evaluate mutation only variant
  • Evaluate mutation tournament variant
  • Result:
  • Best performance for canonical MOMIES
  • Step size in continuous and integer space adapts quite well
  • Chaotic step size behaviour in categorical space
  • Future:
  • Improve categorial step size adaptation
  • Investigate recombination behaviour

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References I

  • [Emmerich et al 2005] M. Emmerich, N. Beume, and B. Naujoks, “An EMO algorithm using

the hypervolume measure as selection criterion,” in Evolutionary Multi-Criterion Optimization, edited by C. A. Coello Coello, A. Hernández Aguirre, and E. Zitzler (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005), pp. 62–76.

  • [Emmerich+Deutz 2018] M. T. M. Emmerich and A. H. Deutz, Natural Computing 17, 585–

609Sep (2018).

  • [Laredo 2015] D. Laredo Razo, EDS: A Continuation Method for Mixed-Integer Multi-
  • bjective Optimization Problems, Master’s thesis, CINVESTAV-IPN, Mexico City (2015).
  • [Li et al 2013] R. Li, M. T. M. Emmerich, J. Eggermont, T. Bäck, M. Schütz, J. Dijkstra, and J.
  • H. C. Reiber, Evolutionary computation 21, 29–64 (2013).
  • [Przybylski 2010] A. Przybylski, X. Gandibleux, and M. Ehrgott, INFORMS Journal on

Computing 22, 371–386 (2010).

  • [Rudolph 1994] G. Rudolph, “An evolutionary algorithm for integer programming,” in

Parallel Problem Solving from Nature — PPSN III, edited by Y. Davidor, H.-P. Schwefel, and

  • R. Männer (Springer Berlin Heidelberg, Berlin, Heidelberg, 1994), pp. 139–148.
  • [Schwefel 1981] Hans-Paul Schwefel. Numerical Optimization of Computer Models. John

Wiley & Sons, Inc., New York, NY, USA, 1981.

  • [Wang 2015] H. Wang, Computers & Operations Research 61, 100–09 (2015).

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References II

  • [Wessing et al 2017] S. Wessing, R. Pink, K. Brandenbusch, and G. Rudolph, “Toward step-

size adaptation in evolutionary multi-objective optimization,” in Evolutionary Multi- Criterion Optimization, edited by H. Trautmann, G. Rudolph, K. Klamroth, O. Schütze, M. Wiecek, Y. Jin, and C. Grimme (Springer International Publishing, Cham, 2017), pp. 670– 684. 23

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Sphere convergence

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Step size adaptation

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