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Basin Hopping Networks of continuous global optimization problems

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Abstract

Characterization of optimization problems with respect to their solvability is one of the focal points of many research projects in the field of global optimization. Our study contributes to these efforts with the usage of the computational and mathematical tools of network science. Given an optimization problem, a network formed by all the minima found by an optimization method can be constructed. In this paper we use the Basin Hopping method on well-known benchmarking problems and investigate the resulting networks using several measures.

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Notes

  1. Note that the optimization problem (1) can also be extended to have constraints, although in the experimental part of our paper we will investigate only box-constrained problems of form (1).

References

  • Ackley D (2012) A connectionist machine for genetic hillclimbing, vol 28. Springer, Berlin

    Google Scholar 

  • Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97

    Article  Google Scholar 

  • Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exp 10:P10008

    Article  Google Scholar 

  • Brin S, Page L (1998) The anatomy of a large-scale hypertextual web search engine. Comput Netw 30:107–117

    Google Scholar 

  • Cho H, Olivera F, Guikema SD (2008) A derivation of the number of minima of the Griewank function. Appl Math Comput 204:694–701

    Google Scholar 

  • Daolio F, Tomassini M, Vérel S, Ochoa G (2011) Communities of minima in local optima networks of combinatorial spaces. Phys A Stat Mech Appl 390:1684–1694

    Article  Google Scholar 

  • Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201–213

    Article  Google Scholar 

  • Doye JPK (2002) The network topology of a potential energy landscape: a static scale-free network. Phys Rev Lett 88:238701

    Article  Google Scholar 

  • Erdős P, Rényi A (1959) On random graphs. Publ Math 6:290–297

    Google Scholar 

  • Fourer R, Kernighan BW (2002) AMPL: a modeling language for mathematical programming. Duxbury Press, North Scituate

    Google Scholar 

  • Freeman L (1977) A set of measures of centrality based on betweenness. Sociometry 40:35–41

    Article  Google Scholar 

  • Griewank AO (1981) Generalized descent for global optimization. J Optim Theory Appl 34:11–39

    Article  Google Scholar 

  • Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680

    Article  Google Scholar 

  • Leary RH (2000) Global optimization on funneling landscapes. J Glob Optim 18:367–383

    Article  Google Scholar 

  • Levy AV, Montalvo A, Gomez S, Calderon A (1981) Topics in global optimization, Lecture Notes in Mathematics, No. 909, Springer-Verlag, Berlin

  • Locatelli M (2003) A note on the Griewank Test Function. J Glob Optim 25:169–174

    Article  Google Scholar 

  • Locatelli M (2005) On the multilevel structure of global optimization problems. Comput Optim Appl 30:5–22

    Article  Google Scholar 

  • Locatelli M, Schoen F (2013) Global optimization: theory. Algorithms and applications. SIAM-MOS, Philadelphia

    Book  Google Scholar 

  • Locatelli M, Maischberger M, Schoen F (2014) Differential evolution methods based on local searches. Comput Oper Res 43:169–180

    Article  Google Scholar 

  • Mittelmann H (2017) Benchmarks for optimization software. http://plato.asu.edu/bench.html. Accessed 17 Jan 2017

  • More JJ, Munson TS (2002) Computing mountain passes, Preprint ANL/MCS-P957- 0502

  • Murtagh BA, Saunders MA (2003) MINOS 5.51 user’s guide, Technical Report SOL 83-20R

  • Neumaier A (2004) Complete search in continuous global optimization and constraint satisfaction. Acta Numer 13:271–369

    Article  Google Scholar 

  • Neumaier A, Shcherbina O, Huyer W, Vinkó T (2005) A comparison of complete global optimization solvers. Math Program 103:335–356

    Article  Google Scholar 

  • Newman MEJ (2010) Networks: an introduction. Oxford University Press, Oxford

    Book  Google Scholar 

  • Newman MEJ (2005) A measure of betweenness centrality based on random walks. Soc Netw 27:39–54

    Article  Google Scholar 

  • Newman MEJ (2006) Modularity and community structure in networks. PNAS 103:8577–8696

    Article  Google Scholar 

  • Radicchi F, Castellano C, Cecconi F, Loreto V, Parisi D (2004) Defining and identifying communities in networks. PNAS 101(9):2658–2663

    Article  Google Scholar 

  • Rios LM, Sahinidis NV (2013) Derivative-free optimization: a review of algorithms and comparison of software implementations. J Glob Optim 56:1247–1293

    Article  Google Scholar 

  • Scala A, Amaral L, Barthélémy M (2001) Small-world networks and the conformation space of lattice polymer chains. Europhys Lett 55:594–600

    Article  Google Scholar 

  • Stillinger FH, Weber TA (1984) Packing structures and transitions in liquids and solids. Science 225:983–989

    Article  Google Scholar 

  • Storn R, Price K (1997) Differential evolution—a simple and efficient heuristic for global optimisation over continuous spaces. J Glob Optim 11:341–359

    Article  Google Scholar 

  • Suganthan PN, Hansen N, Liang JJ, Deb K, Chen YP, Auger A, Tiwari S (2005) Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization. KanGAL report, 2005005:

  • Tomassini M, Verel S, Ochoa G (2008) Complex-network analysis of combinatorial spaces: the NK landscape case. Phys Rev E 78(6):066114

    Article  Google Scholar 

  • Torn A, Zilinskas A (1989) Global optimization. Springer, New York

    Book  Google Scholar 

  • Wales DJ, Doye JPK (1997) Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms. J Phys Chem A 101:5111–5116

    Article  Google Scholar 

  • Zabinsky ZB, Smith RL (1992) Pure adaptive search in global optimization. Math Program 53:323–338

    Article  Google Scholar 

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Acknowledgements

The authors would like o thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. T. Vinkó was supported by the Bolyai Scholarship of the Hungarian Academy of Sciences.

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Authors and Affiliations

  1. Institute of Informatics, University of Szeged, Árpád tér 2, Szeged, 6720, Hungary

    Tamás Vinkó & Kitti Gelle

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  1. Tamás Vinkó
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  2. Kitti Gelle
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Correspondence to Tamás Vinkó.

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Vinkó, T., Gelle, K. Basin Hopping Networks of continuous global optimization problems. Cent Eur J Oper Res 25, 985–1006 (2017). https://doi.org/10.1007/s10100-017-0480-0

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