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On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach

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  • Benjamin Martin
  • Alexandre Goldsztejn
  • Laurent Granvilliers
  • Christophe Jermann

Abstract

The global resolution of constrained non-linear bi-objective optimization problems (NLBOO) aims at covering their Pareto-optimal front which is in general a one-manifold in $$\mathbb {R}^2$$ R 2 . Continuation methods can help in this context as they can follow a continuous component of this front once an initial point on it is provided. They constitute somehow a generalization of the classical scalarization framework which transforms the bi-objective problem into a parametric single-objective problem. Recent works have shown that they can play a key role in global algorithms dedicated to bi-objective problems, e.g. population based algorithms, where they allow discovering large portions of locally Pareto optimal vectors, which turns out to strongly support diversification. The contribution of this paper is twofold: we first provide a survey on continuation techniques in global optimization methods for NLBOO, identifying relations between several work and usual limitations, among which the ability to handle inequality constraints. We then propose a rigorous active set management strategy on top of a continuation method based on interval analysis, certified with respect to feasibility, local optimality and connectivity. This allows overcoming the latter limitation as illustrated on a representative bi-objective problem. Copyright Springer Science+Business Media New York 2016

Suggested Citation

  • Benjamin Martin & Alexandre Goldsztejn & Laurent Granvilliers & Christophe Jermann, 2016. "On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach," Journal of Global Optimization, Springer, vol. 64(1), pages 3-16, January.
    • Handle: RePEc:spr:jglopt:v:64:y:2016:i:1:p:3-16
    DOI: 10.1007/s10898-014-0201-3
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    References listed on IDEAS

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    Cited by:

    1. Charles Audet & Frédéric Messine & Jordan Ninin, 2022. "Numerical certification of Pareto optimality for biobjective nonlinear problems," Journal of Global Optimization, Springer, vol. 83(4), pages 891-908, August.
    2. Ignacio Araya & Jose Campusano & Damir Aliquintui, 2019. "Nonlinear biobjective optimization: improvements to interval branch & bound algorithms," Journal of Global Optimization, Springer, vol. 75(1), pages 91-110, September.
    3. Martin, Benjamin & Goldsztejn, Alexandre & Granvilliers, Laurent & Jermann, Christophe, 2017. "Constraint propagation using dominance in interval Branch & Bound for nonlinear biobjective optimization," European Journal of Operational Research, Elsevier, vol. 260(3), pages 934-948.
    4. Li, Mingwu & Dankowicz, Harry, 2020. "Optimization with equality and inequality constraints using parameter continuation," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    5. Ricardo Landa & Giomara Lárraga & Gregorio Toscano, 2019. "Use of a goal-constraint-based approach for finding the region of interest in multi-objective problems," Journal of Heuristics, Springer, vol. 25(1), pages 107-139, February.
    6. Ignacio Araya & Damir Aliquintui & Franco Ardiles & Braulio Lobo, 2021. "Nonlinear biobjective optimization: improving the upper envelope using feasible line segments," Journal of Global Optimization, Springer, vol. 79(2), pages 503-520, February.
    7. Lourdes Uribe & Johan M Bogoya & Andrés Vargas & Adriana Lara & Günter Rudolph & Oliver Schütze, 2020. "A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems," Mathematics, MDPI, vol. 8(10), pages 1-29, October.

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