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PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization

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Abstract

We introduce a novel approximation method for multiobjective optimization problems called PAINT–SiCon. The method can construct consistent parametric representations of Pareto sets, especially for nonconvex problems, by interpolating between nondominated solutions of a given sampling both in the decision and objective space. The proposed method is especially advantageous in computationally expensive cases, since the parametric representation of the Pareto set can be used as an inexpensive surrogate for the original problem during the decision making process.

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Notes

  1. For a comprehensive survey on well established multiobjective optimization algorithms see [1] and the references therein.

  2. Further technical details on the PAINT algorithm are given in Sect. 2 of the Supplementary Material (SM).

  3. The Pareto critical set is the set of points satisfying the first order necessary conditions for optimality (see [24]).

  4. By generic property, we mean a property that holds for an open and dense set, or more generally on a residual set, i.e., a set which is a countable intersection of open and dense sets. Notice that the sets we are dealing with are function spaces, i.e., we are saying that a generic property holds for “almost all” functions in a suitable sense. See, among others, [24, 30] and the references therein.

  5. A stratified set can be thought as a differentiable manifold with boundary, where the boundary is a collection of differentiable manifolds of lower dimension, and the same holds for the boundary of those manifolds. See the SM for more details.

  6. A tubular neighborhood of a set \(S\subset X\) with radius \(r>0\) is defined as \(T_{S,r}:=\left\{ x\in X: d(x,S)<r\right\} \).

  7. See also Fig. 3b.

  8. See also Fig. 4.

  9. Note that, all the simplexes are now line segments.

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Acknowledgments

This research has been partially supported by the Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”.

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Authors and Affiliations

  1. Department of Mathematical Information Technology, University of Jyväskylä, PO Box 35 (Agora), 40014, Jyväskylä, Finland

    Markus E. Hartikainen

  2. Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63, 35121, Padova, Italy

    Alberto Lovison

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  1. Markus E. Hartikainen
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  2. Alberto Lovison
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Correspondence to Alberto Lovison.

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Hartikainen, M.E., Lovison, A. PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization. J Glob Optim 62, 243–261 (2015). https://doi.org/10.1007/s10898-014-0232-9

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