Abstract
An important issue in multiobjective optimization is the study of the convergence speed of algorithms. An optimization problem must be defined as simple as possible to minimize the computational cost required to solve it. In this work, we study the convergence speed of seven multiobjective evolutionary algorithms: DEPT, MO-VNS, MOABC, MO-GSA, MO-FA, NSGA-II, and SPEA2; when solving an important biological problem: the motif discovery problem. We have used twelve instances of four different organisms as benchmark, analyzing the number of fitness function evaluations required by each algorithm to achieve reasonable quality solutions. We have used the hypervolume indicator to evaluate the solutions discovered by each algorithm, measuring its quality every 100 evaluations. This methodology also allows us to study the hit rates of the algorithms over 30 independent runs. Moreover, we have made a deeper study in the more complex instance of each organism. In this study, we observe the increase of the archive (number of non-dominated solutions) and the spread of the Pareto fronts obtained by the algorithm in the median execution. As we will see, our study reveals that DEPT, MOABC, and MO-FA provide the best convergence speeds and the highest hit rates.
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Ao W, Gaudet J, Kent WJ, Muttumu S, Mango SE (2004) Environmentally induced foregut remodeling by PHA-4/FoxA and DAF-12/NHR. Science 305(5691):1743–1746
Bailey TL, Elkan C (1995) Unsupervised learning of multiple motifs in biopolymers using expectation maximization. Mach Learn 21(1–2):51–80
Bechikh S, Ben Said L, Ghédira K (2010) Estimating nadir point in multi-objective optimization using mobile reference points. IEEE congress on, evolutionary computation (CEC’10), pp 1–9
Ben Said L, Bechikh S, Ghédira K (2010) The r-dominance: a new dominance relation for interactive evolutionary multicriteria decision making. IEEE Trans Evol Comput 14(5):801–818
Boskovic B, Zumer V (2010) An improved self-adaptive differential evolution algorithm in single objective constrained real-parameter optimization. IEEE congress on, evolutionary computation (CEC’10), pp 1–8
Burset M, Guig R (1996) Evaluation of gene structure prediction programs. Genomics 34(3):353–367
Che D, Song Y, Rashedd K (2005) MDGA: motif discovery using a genetic algorithm. Proceedings of the 2005 conference on genetic and, evolutionary computation (GECCO’05), pp 447–452
Crooks GE, Hon G, Chandonia JM, Brenner SE (2004) WebLogo: a sequence logo generator. Genome Res 14:1188–1190
Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, New York
Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6:182–197
D’haeseleer P (2006) What are DNA sequence motifs? Nature Biotechnol 24(4):423–425
Eskin E, Pevzner PA (2002) Finding composite regulatory patterns in DNA sequences. Bioinformatics 18(Suppl 1):S354–S363
Favorov AV, Gelfand MS, Gerasimova AV, Ravcheev DA, Mironov AA, Makeev VJ (2005) A Gibbs sampler for identification of symmetrically structured, spaced DNA motifs with improved estimation of the signal length. Bioinformatics 21(10):2240–2245
Fogel GB, Porto VW, Varga G, Dow ER, Crave AM, Powers DM, Harlow HB, Su EW, Onyia JE, Su C (2008) Evolutionary computation for discovery of composite transcription factor binding sites. Nucl Acids Res 36(21):e142, 1–14
Fogel GB, Weekes DG, Varga G, Dow ER, Harlow HB, Onyia JE, Su C (2004) Discovery of sequence motifs related to coexpression of genes using evolutionary computation. Nucl Acids Res 32(13):3826–3835
Frith MC, Hansen U, Spouge JL, Weng Z (2004) Finding functional sequence elements by multiple local alignment. Nucl Acids Res 32(1):189–200
González-Álvarez DL, Vega-Rodríguez MA, Gómez-Pulido JA, Sánchez-Pérez JM (2010) A multiobjective variable neighborhood search for solving the motif discovery problem. International Workshop on Soft Computing Models in Industrial Applications (SOCO’10) vol 73, pp 39–46
González-Álvarez DL, Vega-Rodríguez MA, Gómez-Pulido JA, Sánchez-Pérez JM (2010) Solving the motif discovery problem by using differential evolution with pareto tournaments. Proceedings of the 2010 IEEE congress on, evolutionary computation (CEC’10), pp 4140–4147
González-Álvarez DL, Vega-Rodríguez MA, Gómez-Pulido JA, Sánchez-Pérez JM (2011) Applying a multiobjective gravitational search algorithm (MO-GSA) to discover motifs. International work conference on artificial neural networks (IWANN’11), LNCS 6692/2011:372–379
González-Álvarez DL, Vega-Rodríguez MA, Gómez-Pulido JA, Sánchez-Pérez JM (2011) Finding motifs in DNA sequences applying a multiobjective artificial bee colony (MOABC) algorithm. Evolutionary computation, machine learning and data mining in bioinformatics. Lect Notes Comput Sci 6623(2011):89–100
González-Álvarez DL, Vega-Rodríguez MA, Gómez-Pulido JA, Sánchez-Pérez JM (2011) Predicting DNA motifs by using evolutionary multiobjective optimization. IEEE Trans Syst Man Cybern Part C Appl Rev 42(6):913–925
González-Álvarez DL, Vega-Rodríguez MA, Gómez-Pulido JA, Sánchez-Pérez JM (2012) Comparing multiobjective artificial bee colony adaptations for discovering DNA motifs. Evolutionary computation, machine learning and data mining in bioinformatics. Lect Notes Comput Sci 7246:110–121
González-Álvarez DL, Vega-Rodríguez MA, Gómez-Pulido JA, Sánchez-Pérez JM (2012) Comparing multiobjective swarm intelligence metaheuristics for DNA motif discovery. Eng Appl Artif Intell 26(1):314–326
Hertz GZ, Stormo GD (1999) Identifying DNA and protein patterns with statistically significant alignments of multiple sequences. Bioinformatics 15(7–8):563–577
Karaboga D (2005) An idea based on honey bee swarm for numerical optimization. Technical report-tr06, Erciyes University, Turkey.
Kaya M (2009) MOGAMOD: Multi-objective genetic algorithm for motif discovery. Expert Syst Appl 36(2):1039–1047
Li L (2009) GADEM: A genetic algorithm guided formation of spaced dyads coupled with an EM algorithm for motif discovery. J Computat Biol 16(2):317–329
Liu FFM, Tsai JJP, Chen RM, Chen SN, Shih SH (2004) FMGA: finding motifs by genetic algorithm. Fourth IEEE symposium on bioinformatics and, bioengineering (BIBE’04), pp 459–466
Lones MA, Tyrrell AM (2007) Regulatory motif discovery using a population clustering evolutionary algorithm. IEEE/ACM Trans Computat Biol Bioinf 4(3):403–414
Mladenovic N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24:1097–1100
Pavesi G, Mauri G, Pesole G (2001) An algorithm for finding signals of unknown length in DNA sequences. Bioinformatics 17(Suppl 1):S207–S214
Pevzner P, Sze SH (2000) Combinatorial approaches to finding subtle signals in dna sequences. Proceedings of the eighth international conference on intelligent systems for, molecular biology, pp 269–278
Rashedi E, Nezamabadi-pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248
Regnier M, Denise A (2004) Rare events and conditional events on random strings. Discr Math Theoret Comput Sci 6:191–214
Roth FP, Hughes JD, Estep PW, Church GM (1998) Finding DNA regulatory motifs within unaligned noncoding sequences clustered by whole-genome mRNA quantitation. Nature Biotechnol 16(10):939–945
Shao L, Chen Y (2009) Bacterial foraging optimization algorithm integrating tabu search for motif discovery. IEEE international conference on bioinformatics and, biomedicine (BIBM’09), pp 415- 418
Shao L, Chen Y, Abraham A (2009) Motif discovery using evolutionary algorithms. International conference of soft computing and, pattern recognition (SOCPAR’09) pp 420–425
Sheskin DJ (2007) Handbook of parametric and nonparametric statistical procedures, 4th edn. Chapman & Hall/CRC Press, New York
Sinha S, Tompa M (2003) YMF: a program for discovery of novel transcription factor binding sites by statistical overrepresentation. Nucl Acids Res 31(13):3586–3588
Stine M, Dasgupta D, Mukatira S (2003) Motif discovery in upstream sequences of coordinately expressed genes. The 2003 congress on, evolutionary computation (CEC’03) vol 3, pp 1596–1603
Storn R, Price K (1997) Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341–359
Talbi EG (2009) Metaheuristics: from design to implementation. Wiley, New York
Thijs G, Lescot M, Marchal K, Rombauts S, De Moor B, Rouzé P, Moreau Y (2001) A higher-order background model improves the detection of promoter regulatory elements by Gibbs sampling. Bioinformatics 17(12):1113–1122
Tompa M, Li N, Bailey TL, Church GM, De Moor B, Eskin E, Favorov AV, Frith MC, Fu Y, Kent WJ, Makeev VJ, Mironov AA, Noble WS, Pavesi G, Pesole G, Rgnier M, Simonis N, Van Helden J, Vandenbogaert M, Weng Z, Workman C, Ye C, Zhu Z (2005) Assessing computational tools for the discovery of transcription factor binding sites. Nature Biotechnology 23(1):137–144
van Helden J, Andre B, Collado-Vides J (1998) Extracting regulatory sites from the upstream region of yeast genes by computational analysis of oligonucleotide frequencies. J Mol Biol 281(5):827–842
van Helden J, Rios AF, Collado-Vides J (2000) Discovering regulatory elements in non-coding sequences by analysis of spaced dyads. Nucl Acids Res 28(8):1808–1818
Wagner T, Beume N, Naujoks B (2007) Pareto-, aggregation-, and indicator-based methods in manyobjective optimization. Evolutionary multi-criterion optimization. Lect Notes Comput Sci 4403:742–756
Wingender E, Dietze P, Karas H, Knuppel R (1996) TRANSFAC: a database on transcription factors and their DNA binding sites. Nucl Acids Res 24(1):238–241
Workman CT, Stormo GD (2000) ANN-Spec: a method for discovering transcription factor binding sites with improved specificity. Pacific symposium on biocomputing, pp 467–478
Yang XS (2009) Firefly algorithms for multimodal optimization. 5th international symposium of stochastic algorithms: foundations and applications (SAGA’09). LNCS 5792:169–178
Zhang Q, Li H (2007) MOEA/D: a multi-objective evolutionary algorithm based on decomposition. IEEE Trans Evol Computat 11(6):712–731
Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans Evol Comput 3(4):257–271
Zitzler E, Kunzli S (2004) Indicator-based selection in multiobjective search. Proceedings 8th international conference on parallel problem solving from nature (PPSN VIII), pp 832–842
Zitzler E, Laumanns M, Thiele L (2002) SPEA2: improving the strength pareto evolutionary algorithm. International conference on evolutionary and deterministic methods for design, optimization and control with applications in industrial problems (EUROGEN), pp 95–100
Acknowledgments
This work was partially funded by the Spanish Ministry of Economy and Competitiveness and the ERDF (European Regional Development Fund), under the contract TIN2012-30685 (BIO project). Thanks also to the Fundación Valhondo for the economic support offered to David L. González-Álvarez. Álvaro Rubio-Largo is supported by the research Grant PRE09010 from Gobierno de Extremadura (Spain) and the European Social Fund (ESF).
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Communicated by G. Acampora.
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González-Álvarez, D.L., Vega-Rodríguez, M.A. & Rubio-Largo, Á. Convergence analysis of some multiobjective evolutionary algorithms when discovering motifs. Soft Comput 18, 853–869 (2014). https://doi.org/10.1007/s00500-013-1103-x
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DOI: https://doi.org/10.1007/s00500-013-1103-x