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A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
A Pareto-Compliant Surrogate Approach  for Multiobjective Optimization
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A Pareto-Compliant Surrogate Approach for Multiobjective Optimization

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Most surrogate approaches to multi-objective optimization …

Most surrogate approaches to multi-objective optimization
build a surrogate model for each objective. These surrogates
can be used inside a classical Evolutionary Multiobjective
Optimization Algorithm (EMOA) in lieu of the actual objectives, without modifying the underlying EMOA; or to filter
out points that the models predict to be uninteresting. In
contrast, the proposed approach aims at building a global
surrogate model defined on the decision space and tightly
characterizing the current Pareto set and the dominated region, in order to speed up the evolution progress toward
the true Pareto set. This surrogate model is specified by
combining a One-class Support Vector Machine (SVMs) to
characterize the dominated points, and a Regression SVM to
clamp the Pareto front on a single value. The resulting surrogate model is then used within state-of-the-art EMOAs to
pre-screen the individuals generated by application of standard variation operators. Empirical validation on classical
MOO benchmark problems shows a significant reduction of
the number of evaluations of the actual objective functions.

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  • 1. A Pareto-Compliant Surrogate Approach for Multiobjective Optimization Ilya Loshchilov, Marc Schoenauer and Michéle Sebag TAO, INRIA, Univesité Paris-Sud XI
  • 2. Main Idea → Pareto Surrogate WE HAVE: • Current Pareto and dominated points WE WANT TO FIND F: decision space → • F < 0 for dominated points • F ≈ 0 for current Pareto points WE EXPECT: • F > 0 for new Pareto pointsA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 2
  • 3. Outline Support Vector Machine (SVM) A Pareto-Compliant Approach for MOO Experiments Discussion and on-going workA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 3
  • 4. Support Vector Machine (1): ClassificationA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 4
  • 5. Support Vector Machine (2): RegressionA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 5
  • 6. Support Vector Machine (3): One-Class SVMA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 6
  • 7. Outline Support Vector Machine (SVM) A Pareto-Compliant Approach for MOO Experiments Discussion and on-going workA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 7
  • 8. Pareto Support Vector Machine (Pareto-SVM)A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 8
  • 9. SVM-informed EMOA How we can use F? For optimization • If we are sure that the model F is accurate enough far from the current Pareto points For filtering (select the most promising children according to F): • F(children) > F(parent) → dangerous because of approximation noise (epsilon) • F(children) > F(nearest point from population) is more robust criterionA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 9
  • 10. SVM-informed EMOA: Filtering Filtering procedure: Generate pre-children For each pre-children and the nearest parent calculate New children is the point with the maximum value ofA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 10
  • 11. Outline Support Vector Machine (SVM) A Pareto-Compliant Approach for MOO Experiments Discussion and on-going workA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 11
  • 12. Pareto-SVM model validation Generate 20.000 points at a given distance from a (known) nearly-optimal Pareto front Apply Non-dominated sorting to rank non-dominated fronts , where Choose and as the training data, where – current Pareto points, – dominated points Update Pareto-SVM model Compare on different (use as the Test data)A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 12
  • 13. Pareto-SVM model for ZDT1 and IHR1 problems A, b,c,dA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 13
  • 14. SVM-informed S-NSGA-2 and MO-CMA-ES on ZDTs and IHRsA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 14
  • 15. Strong and Weak Points + Speedup of 2-2.5 on ZDT and IHR for first 15 000 evaluations + CPU cost of one learning ~ 0.5-1.0 sec. (complete run<15mn) - High selection pressure may lead to premature convergence; - The formulation of the SVM optimization problem still requires the user decision*.A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 15
  • 16. Issue: A more robust Pareto-SVM formulation Original Formulation: In feature space: - Pareto set → cluster around single line - Dominated points → one side of this line Issues: - Which side? - Preserve symmetry ( and ) Solutions(?): - Manually choose - New Formulation: D: additional parameter → harder learning problemA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 16
  • 17. Perspective (1): Quality indicator-based Mono surrogate Regression of indicator values (e.g. hypervolume) Issues: - value for dominated points? - value for extremal points?A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 17
  • 18. Perspective (2): Rank-SVM (Ordinal Regression SVM) Ordinal Regression SVM: surrogate constrained by ordered pairs MO-case: derive constraints from Pareto dominance If X dominates Y, add constraint (F(X)<F(Y)) Learning time - quadratic with number of constraints - quasi-linear with dimension Which constraints? Results for single objective problems(*): Cost of learning/testing with n-1 constraints increases quasi-linearly with dimension. (*) - "Comparison-Based Optimizers Need Comparison-Based Surrogates” I.Loshchilov, M. Schoenauer, M. Sebag. In PPSN-XI (2010).A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 18
  • 19. Summary Contributions Single-valued surrogate to capture Pareto dominance Several alternative formulations Using surrogate as filter limits possible speedup Further work Comparison with multi-surrogate Rank-based SVMA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 19
  • 20. Thank youA Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 20

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