ABSTRACT
Complex problems are often addressed by methods from the domain of computational intelligence, including metaheuristic algorithms. Different metaheuristics have different abilities to solve specific types of problems and the selection of suitable methods has a large impact on the ability to find good problem solutions. Problem characterization became an important step in the application of intelligent methods to practical problems. A popular approach to problem characterization is the exploratory landscape analysis. It consists of a sequence of operations that approximate and describe the hypersurfaces formed by characteristic problem properties from a limited sample of solutions. Exploratory landscape analysis uses a particular strategy to select just a small subset of problem solutions for which are the characteristic properties evaluated and high-level landscape features computed. Low-discrepancy sequences have been recently used to design a family of sampling strategies. They have useful space-filling properties but their effective and efficient randomization might represent an issue. In this work, we study the Cranley-Patterson rotation, a lightweight randomization strategy for low-discrepancy sequences, compare it with other randomization methods, and observe the effect its use has on the randomization of sets of sampling points in the context of exploratory landscape analysis.
- Abdalla G.M. Ahmed, Hélène Perrier, David Coeurjolly, Victor Ostromoukhov, Jianwei Guo, Dong-Ming Yan, Hui Huang, and Oliver Deussen. 2016. Low-Discrepancy Blue Noise Sampling. ACM Transactions on Graphics 35, 6 (2016), 247:1–247:13. https://doi.org/10.1145/2980179.2980218Google ScholarDigital Library
- Emanouil Atanassov, Sofiya Ivanovska, and Aneta Karaivanova. 2021. Optimization of the Direction Numbers of the Sobol Sequences. In Advances in High Performance Computing, Ivan Dimov and Stefka Fidanova (Eds.). Springer International Publishing, Cham, 145–154.Google Scholar
- Emanouil Atanassov, Aneta Karaivanova, and Sofiya Ivanovska. 2010. Tuning the Generation of Sobol Sequence with Owen Scrambling. In Large-Scale Scientific Computing, Ivan Lirkov, Svetozar Margenov, and Jerzy Waśniewski (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 459–466.Google Scholar
- H. Chi, M. Mascagni, and T. Warnock. 2005. On the Optimal Halton Sequence. Mathematics and Computers in Simulation 70, 1 (2005), 9–21. https://doi.org/10.1016/j.matcom.2005.03.004Google ScholarDigital Library
- François-Michel De Rainville, Christian Gagné, Olivier Teytaud, and Denis Laurendeau. 2009. Optimizing Low-discrepancy Sequences with an Evolutionary Algorithm. In Proceedings of the 11th Annual Conference on Genetic and Evolutionary Computation (Montreal, Quebec, Canada) (GECCO ’09). ACM, New York, NY, USA, 1491–1498. https://doi.org/10.1145/1569901.1570101Google ScholarDigital Library
- François-Michel De Rainville, Christian Gagné, Olivier Teytaud, and Denis Laurendeau. 2012. Evolutionary Optimization of Low-discrepancy Sequences. ACM Trans. Model. Comput. Simul. 22, 2, Article 9 (March 2012), 25 pages. https://doi.org/10.1145/2133390.2133393Google ScholarDigital Library
- Josef Dick, Frances Y. Kuo, and Ian H. Sloan. 2013. High-dimensional integration: The quasi-Monte Carlo way. Acta Numerica 22 (2013), 133–288. https://doi.org/10.1017/S0962492913000044Google ScholarCross Ref
- Carola Doerr and François-Michel De Rainville. 2013. Constructing Low Star Discrepancy Point Sets with Genetic Algorithms. In Proc. of the 15th Annual Conf. on Genetic and Evolutionary Computation (Amsterdam, The Netherlands) (GECCO ’13). ACM, New York, NY, USA, 789–796. https://doi.org/10.1145/2463372.2463469Google ScholarDigital Library
- John H Halton. 1964. Algorithm 247: Radical-inverse quasi-random point sequence. Commun. ACM 7, 12 (1964), 701–702.Google ScholarDigital Library
- R.V. Hogg and E.A. Tanis. 2006. Probability and Statistical Inference. Prentice Hall.Google Scholar
- P. Kromer, V. Uher, A. Andova, T. Tusar, and B. Filipic. 2022. Sampling Strategies for Exploratory Landscape Analysis of Bi-Objective Problems. In 2022 International Conference on Computational Science and Computational Intelligence (CSCI). IEEE Computer Society, Los Alamitos, CA, USA, 336–342. https://doi.org/10.1109/CSCI58124.2022.00067Google ScholarCross Ref
- P. Krömer, J. Platoš, and V. Snášel. 2020. Differential evolution for the optimization of low-discrepancy generalized Halton sequences. Swarm and Evolutionary Computation 54 (2020), 100649. https://doi.org/10.1016/j.swevo.2020.100649Google ScholarCross Ref
- Pierre L’Ecuyer. 2018. Randomized Quasi-Monte Carlo: An Introduction for Practitioners. In Monte Carlo and Quasi-Monte Carlo Methods, Art B. Owen and Peter W. Glynn (Eds.). Springer International Publishing, Cham, 29–52.Google Scholar
- Pierre L’Ecuyer and Christiane Lemieux. 2000. Variance Reduction via Lattice Rules. Management Science 46, 9 (2000), 1214–1235.Google ScholarDigital Library
- Christiane Lemieux. 2009. Monte Carlo and Quasi-Monte Carlo Sampling. Springer.Google Scholar
- Yaxin Li, Jing Liang, Kunjie Yu, Ke Chen, Yinan Guo, Caitong Yue, and Leiyu Zhang. 2022. Adaptive local landscape feature vector for problem classification and algorithm selection. Applied Soft Computing 131 (2022), 109751. https://doi.org/10.1016/j.asoc.2022.109751Google ScholarDigital Library
- Katherine Mary Malan. 2021. A Survey of Advances in Landscape Analysis for Optimisation. Algorithms 14, 2 (Feb. 2021), 40. https://doi.org/10.3390/a14020040Google ScholarCross Ref
- Katherine M. Malan and I. Moser. 2019. Constraint Handling Guided by Landscape Analysis in Combinatorial and Continuous Search Spaces. Evolutionary Computation 27, 2 (2019), 267–289. https://doi.org/10.1162/evco_a_00222Google ScholarDigital Library
- Michael D McKay, Richard J Beckman, and William J Conover. 2000. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42, 1 (2000), 55–61.Google ScholarCross Ref
- Olaf Mersmann, Bernd Bischl, Heike Trautmann, Mike Preuss, Claus Weihs, and Günter Rudolph. 2011. Exploratory landscape analysis. In Proceedings of the 13th Annual Genetic and Evolutionary Computation Conference (GECCO). ACM, 829–836.Google ScholarDigital Library
- Art B. Owen. 1995. Randomly Permuted (t,m,s)-Nets and (t, s)-Sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Harald Niederreiter and Peter Jau-Shyong Shiue (Eds.). Springer New York, New York, NY, 299–317.Google Scholar
- Art B. Owen. 2022. On Dropping the First Sobol’ Point. In Monte Carlo and Quasi-Monte Carlo Methods, Alexander Keller (Ed.). Springer International Publishing, Cham, 71–86.Google Scholar
- Igor Radović, Ilya M. Sobol’, and Robert F. Tichy. 1996. Quasi-Monte Carlo Methods for Numerical Integration: Comparison of Different Low Discrepancy Sequences. Monte Carlo Methods and Applications 2, 1 (1996), 1–14. https://doi.org/10.1515/mcma.1996.2.1.1Google ScholarCross Ref
- [24] Quentin Renau, Carola Doerr, Johann Dreo, and Benjamin Doerr. [n. d.].Google Scholar
- Hendrik Richter. 2014. Fitness Landscapes: From Evolutionary Biology to Evolutionary Computation. In Recent Advances in the Theory and Application of Fitness Landscapes. Springer, 3–31.Google ScholarCross Ref
- A. Savine and L. Andersen. 2018. Modern Computational Finance: AAD and Parallel Simulations. Wiley.Google Scholar
- Il’ya Meerovich Sobol. 1967. On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 7, 4 (1967), 784–802.Google Scholar
- Ilya M. Sobol’, Danil Asotsky, Alexander Kreinin, and Sergei Kucherenko. 2011. Construction and Comparison of High-Dimensional Sobol’ Generators. Wilmott 2011, 56 (2011), 64–79. https://doi.org/10.1002/wilm.10056Google ScholarCross Ref
- Xifu Sun, Barry Croke, Stephen Roberts, and Anthony Jakeman. 2021. Comparing methods of randomizing Sobolśequences for improving uncertainty of metrics in variance-based global sensitivity estimation. Reliability Engineering & System Safety 210 (2021), 107499. https://doi.org/10.1016/j.ress.2021.107499Google ScholarCross Ref
- Bart Vandewoestyne and Ronald Cools. 2006. Good permutations for deterministic scrambled Halton sequences in terms of L2-discrepancy. J. Comput. Appl. Math. 189, 1 (2006), 341 – 361. https://doi.org/10.1016/j.cam.2005.05.022Google ScholarDigital Library
- X. Wang and F.J. Hickernell. 2000. Randomized Halton sequences. Mathematical and Computer Modelling 32, 7 (2000), 887–899. https://doi.org/10.1016/S0895-7177(00)00178-3Google ScholarDigital Library
Index Terms
- Randomization of Low-discrepancy Sampling Designs by Cranley-Patterson Rotation
Recommendations
Impact of Different Discrete Sampling Strategies on Fitness Landscape Analysis Based on Histograms
IAIT '23: Proceedings of the 13th International Conference on Advances in Information TechnologyComplex problems are frequently tackled using techniques from the realm of computational intelligence and metaheuristic algorithms. Selection of a metaheuristic from the wide range of algorithms possessing various properties to address specific problem ...
Studying the effect of using low-discrepancy sequences to initialize population-based optimization algorithms
In this paper, we investigate the use of low-discrepancy sequences to generate an initial population for population-based optimization algorithms. Previous studies have found that low-discrepancy sequences generally improve the performance of a ...
Implementation and tests of low-discrepancy sequences
Low-discrepancy sequences are used for numerical integration, in simulation, and in related applications. Techniques for producing such sequences have been proposed by, among others, Halton, Sobol´, Faure, and Niederreiter. Niederreiter's sequences have ...
Comments