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

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 \(\mathbb {R}^2\). 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.

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Notes

  1. 1.

    More precisely, the number of parameters can be reduced to one.

  2. 2.

    A turning point is somehow a U-turn, where the direction of continuation with respect to the fixed parameter locally changes, hence is singular for this parametrization.

  3. 3.

    The activation (or disactivation) of \(k\) constraints at the same time occurs when the one dimensional curve Pareto frontier crosses the intersection of the \(k\) constraints boundaries, which is stable only if \(k=1\).

  4. 4.

    Parallelotopes allow a cheap and certified sampling of the Pareto frontier they include.

    Fig. 3
    figure3

    Captured Pareto-optimal curve in the objective space

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Martin, B., Goldsztejn, A., Granvilliers, L. et al. On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach. J Glob Optim 64, 3–16 (2016). https://doi.org/10.1007/s10898-014-0201-3

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Keywords

  • Non-linear bi-objective optimization
  • Continuation
  • Interval analysis
  • Constraints activity