Nonlinear biobjective optimization: improving the upper envelope using feasible line segments

Abstract

In this work, we propose a segment-based representation for the upper bound of the non-dominated set in interval branch & bound solvers for biobjective non linear optimization. We ensure that every point over the upper line segments is dominated by at least one point in the feasible objective region. Segments are generated by linear envelopes of the image of feasible line segments. Finally, we show that the segment-based representation together with methods for generating upper line segments allows us to converge more quickly to the desired precision of the whole strategy. The code of our solver can be found in our git repository (https://github.com/INFPUCV/ibex-lib/tree/master/plugins/optim-mop).

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Notes

  1. 1.

    Due to the infinite precision of real numbers, computer programs use IEEE-754 floating point numbers [19] for representing reals. Floating point numbers are composed by two integers: a mantissa m and an exponent e which together represent the real number \(m\cdot 2^e\).

  2. 2.

    For the CTP instances, we consider a function \(\phi =1+\sum _{i=2}^n x_i\), otherwise we obtain only one \(\epsilon \)-efficient solution: \(x=(0,0,\ldots )\), \(y=(0,1)\). For the CF3 instances, we consider a size \(n=5\) instead of \(n=2\), because with \(n=2\), we obtain the undetermined term 2/0 in the first objective.

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Acknowledgements

This work is supported by the Fondecyt Project 1200035.

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Correspondence to Ignacio Araya.

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Araya, I., Aliquintui, D., Ardiles, F. et al. Nonlinear biobjective optimization: improving the upper envelope using feasible line segments. J Glob Optim 79, 503–520 (2021). https://doi.org/10.1007/s10898-021-00991-7

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Keywords

  • Interval methods
  • Branch & bound
  • Multiobjective optimization