Abstract
Black-box optimization (BBO) can be used to optimize functions whose analytic form is unknown. A common approach to realising BBO is to learn a surrogate model which approximates the target black-box function which can then be solved via white-box optimization methods. In this paper, we present our approach BOX-QUBO, where the surrogate model is a QUBO matrix. However, unlike in previous state-of-the-art approaches, this matrix is not trained entirely by regression, but mostly by classification between “good” and “bad” solutions. This better accounts for the low capacity of the QUBO matrix, resulting in significantly better solutions overall. We tested our approach against the state-of-the-art on four domains and in all of them BOX-QUBO showed better results. A second contribution of this paper is the idea to also solve white-box problems, i.e. problems which could be directly formulated as QUBO, by means of black-box optimization in order to reduce the size of the QUBOs to the information-theoretic minimum. Experiments show that this significantly improves the results for MAX-k-SAT.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aramon, M., Rosenberg, G., Valiante, E., Miyazawa, T., Tamura, H., Katzgraber, H.G.: Physics-inspired optimization for quadratic unconstrained problems using a digital annealer. Front. Phys. 7, 48 (2019)
Baptista, R., Poloczek, M.: Bayesian optimization of combinatorial structures. In: International Conference on Machine Learning, pp. 462–471. PMLR (2018)
Barahona, F.: On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15(10), 3241 (1982)
Boothby, K., Bunyk, P., Raymond, J., Roy, A.: Technical report: next-generation topology of d-wave quantum processors (2019)
Chancellor, N., Zohren, S., Warburton, P.A., Benjamin, S.C., Roberts, S.: A direct mapping of max k-SAT and high order parity checks to a chimera graph. Sci. Rep. 6, 37107 (2016). https://doi.org/10.1038/srep37107
Choi, V.: Adiabatic quantum algorithms for the NP-complete maximum-weight independent set, exact cover and 3SAT problems. arXiv preprint arXiv:1004.2226 (2010)
De Boer, P.T., Kroese, D.P., Mannor, S., Rubinstein, R.Y.: A tutorial on the cross-entropy method. Ann. Oper. Res. 134(1), 19–67 (2005)
Denchev, V.S., et al.: What is the computational value of finite-range tunneling? Phys. Rev. X 6(3), 031015 (2016)
Huang, T., Goh, S.T., Gopalakrishnan, S., Luo, T., Li, Q., Lau, H.C.: QROSS: QUBO relaxation parameter optimisation via learning solver surrogates. In: 2021 IEEE 41st International Conference on Distributed Computing Systems Workshops (ICDCSW), pp. 35–40. IEEE (2021)
Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355 (1998)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9
Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 220, 671–680 (2000)
Kitai, K., et al.: Designing metamaterials with quantum annealing and factorization machines. Phys. Rev. Res. 2(1), 013319 (2020)
Koshikawa, A.S., Ohzeki, M., Kadowaki, T., Tanaka, K.: Benchmark test of black-box optimization using d-wave quantum annealer. J. Phys. Soc. Jpn. 90(6), 064001 (2021)
Koshikawa, A.S., et al.: Combinatorial black-box optimization for vehicle design problem. arXiv preprint arXiv:2110.00226 (2021)
Lewis, M., Glover, F.: Quadratic unconstrained binary optimization problem preprocessing: theory and empirical analysis. Networks 70(2), 79–97 (2017)
Lodewijks, B.: Mapping NP-hard and NP-complete optimisation problems to quadratic unconstrained binary optimisation problems. arXiv preprint arXiv:1911.08043 (2019)
Lucas, A.: Ising formulations of many NP problems. Front. Phys. 2(5), 1–15 (2014)
Nüßlein, J., Gabor, T., Linnhoff-Popien, C., Feld, S.: Algorithmic QUBO formulations for k-SAT and Hamiltonian cycles. arXiv preprint arXiv:2204.13539 (2022)
Nüßlein, J., Roch, C., Gabor, T., Linnhoff-Popien, C., Feld, S.: Black box optimization using QUBO and the cross entropy method. arXiv preprint arXiv:2206.12510 (2022)
Acknowledgements
This publication was created as part of the Q-Grid project (13N16179) under the “quantum technologies - from basic research to market” funding program, supported by the German Federal Ministry of Education and Research.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Nüßlein, J., Roch, C., Gabor, T., Stein, J., Linnhoff-Popien, C., Feld, S. (2023). Black Box Optimization Using QUBO and the Cross Entropy Method. In: Mikyška, J., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M. (eds) Computational Science – ICCS 2023. ICCS 2023. Lecture Notes in Computer Science, vol 14077. Springer, Cham. https://doi.org/10.1007/978-3-031-36030-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-031-36030-5_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-36029-9
Online ISBN: 978-3-031-36030-5
eBook Packages: Computer ScienceComputer Science (R0)