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Node selection through upper bounding local search methods in branch & bound solvers for NCOPs

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Abstract

Interval-based branch & bound solvers are commonly used for solving Nonlinear Continuous global Optimization Problems (NCOPs). In each iteration, the solver strategically chooses and processes a node within the search tree. The node is bisected and the two generated offspring nodes are processed by filtering methods. For each of these nodes, the solver also searches for new feasible solutions in order to update the best candidate solution. The cost of this solution is used for pruning non-optimal branches of the search tree. Thus, node selection and finding new solutions, stands as pivotal aspects in the functionality of these kind of solvers. The ability to find close-to-optimal solutions early in the search process may discard extensive non-optimal search space regions, thereby effectively reducing the overall size of the search tree. In this work, we propose three novel node selection algorithms that use the feasible solutions obtained through a cost-effective iterative method. Upon updating the best candidate solution, these algorithms strategically choose the node containing this solution for subsequent processing. The newly introduced strategies have been incorporated as node selection methods in a state-of-the-art branch & bound solver, showing promising results in a set of 57 benchmark instances.

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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 \(\varvec{x}_1 \times .. \times \varvec{x}_i \times ... \times \varvec{x}_n\).

  2. https://github.com/vareyesr.

  3. https://arnold-neumaier.at/glopt/coconut/Benchmark/Benchmark.html.

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Acknowledgements

Victor Reyes is supported by Fondecyt Project 11230225. Ignacio Araya is supported by Fondecyt Project 1200035.

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Authors and Affiliations

  1. Universidad Diego Portales, Escuela de Informática y Telecomunicaciones, Santiago, Chile

    Victor Reyes

  2. Pontificia Universidad Católica de Valparaíso, Escuela de Ingeniería Informática, Valparaíso, Chile

    Ignacio Araya

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  1. Victor Reyes
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  2. Ignacio Araya
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Correspondence to Victor Reyes.

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Reyes, V., Araya, I. Node selection through upper bounding local search methods in branch & bound solvers for NCOPs. J Glob Optim (2024). https://doi.org/10.1007/s10898-024-01403-2

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