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Nonlinear biobjective optimization: improvements to interval branch & bound algorithms

Abstract

Interval based solvers are commonly used for solving single-objective nonlinear optimization problems. Their reliability and increasing performance make them useful when proofs of infeasibility and/or certification of solutions are a must. On the other hand, there exist only a few approaches dealing with nonlinear optimization problems, when they consider multiple objectives. In this paper, we propose a new interval branch & bound algorithm for solving nonlinear constrained biobjective optimization problems. Although the general strategy is based on other works, we propose some improvements related to the termination criteria, node selection, upperbounding and discarding boxes using the non-dominated set. Most of these techniques use and/or adapt components of IbexOpt, a state-of-the-art interval-based single-objective optimization algorithm. The code of our plugin can be found in our git repository (https://github.com/INFPUCV/ibex-lib/tree/master/plugins/optim-mop).

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Notes

  1. An interval \({\varvec{x}}_i=[\underline{x_i},\overline{x_i}]\) defines the set of reals \(x_i\), such that \(\underline{x_i} \le x_i \le \overline{x_i}\). A box \({\varvec{x}}\) is a Cartesian product of intervals \({\varvec{x}}_1 \times \cdots \times {\varvec{x}}_i \times \cdots \times {\varvec{x}}_n\).

  2. We implemented, however, the other three simple discarding tests also used in [11], i.e., the dominance test and the simple and generalized monotonicity tests proposed in [8].

  3. For the CTP instances we consider a function \(\varphi =1+\sum \nolimits _{i=2}^n x_i\), otherwise we obtain only one non-dominated solution: \(x=(0,0,...)\), \(y=(0,1)\). 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.

  4. The hypervolume [27] is a quality indicator corresponding to the area of the region dominated by the set of vectors returned by the solver, \(\mathcal {Y}'\), and limited by a reference vector. The optimal hypervolume value corresponds thus to the area of the region dominated by the set of non-dominated vectors \(\mathcal {Y}^*\).

  5. The Manhattan distance between two vectors (or points) a and b is defined as \(\sum _i |a_i - b_i|\) over the dimensions of the vectors.

  6. The larger value obtained for \(N_{max}\) when \(c_y\) is used, can be explained by the fact that, strategies using \(c_y\) generally process a smaller number of boxes (in Table 3, compare the number of processed boxes reported by each strategy), thus it is convenient to generate more feasible solutions in each box.

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Acknowledgements

This work is supported by the Fondecyt Project 1160224.

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Correspondence to Ignacio Araya.

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Araya, I., Campusano, J. & Aliquintui, D. Nonlinear biobjective optimization: improvements to interval branch & bound algorithms. J Glob Optim 75, 91–110 (2019). https://doi.org/10.1007/s10898-019-00768-z

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Keywords

  • Interval methods
  • Branch & bound
  • Multiobjective optimization