Direct search is an important method in numerical optimization. The DIRECT algorithm is a popular global optimization algorithm based on direct search. In this paper, we review firstly a recently proposed multilevel robust DIRECT (MrDIRECT) algorithm, especially how it can eliminate the "eventually inefficient behavior" of the DIRECT algorithm through adopting three levels of search spaces. We provide some more numerical experiments to support MrDIRECT's such ability. Then we propose a new version of MrDIRECT which adopts four levels of search spaces. Our main purpose is to verify the affection of the number of search levels. Extensive numerical results show that four search levels bring no significant improvement but consumes much more computational cost. Therefore, our conclusion is that large number of search levels are not suitable for multilevel search in global optimization. Finally, we pointed out that the idea of multilevel direct search spaces is very convenient in balancing between local search and global search, and therefore can be used to design other global optimization algorithms. Moreover, the numerical result obtained in this paper is helpful for the choice of the number of search levels in multilevel direct search global optimization algorithms.
Androulakis I P,Maranas C D,Floudas C A.αBB:A global optimization method for general constrained nonconvex problems[J].Journal of Global Optimization,1995,7(4):337-363.
Dolan E D,MoréJ J.Benchmarking optimization software with performance profiles[J].Mathematical Programming,2002,91:201-213.
Björkman M,Holmström K.Global optimization using the DIRECT algorithm in Matlab[J].Advanced Modeling and Optimization,1999,1:17-37.
Elsakov S M,Shiryaev V I.Homogeneous algorithms for multiextremal optimization[J].Computational Mathematics and Mathematical Physics,2010,50(10):1642-1654.
Finkel D E.Global optimization with the DIRECT algorithm.PHD thesis,North Carolina State University,2005.
Finkel D E,Kelley C T.Additive scaling and the DIRECT algorithm[J].Journal of Global Optimization,2006,36:597-608.
Floudas C A,Pardalos P M.State of the art in global optimization:computational methods and applications[M].London:Kluwer academic publishers,1996.
Floudas C A,Gounaris C E.A review of recent advances in global optimization[J].Journal of Global Optimization,2009,45:3-38.
Gaviano M,Kvasov D E,Lera D and Sergeyev Ya D.Algorithm 829:Software for Generation of Classes of Test Functions with Known Local and Global Minima for Global Optimization[J].ACM Transactions on Mathematical Software,2003,9(4),469-480.
Gaviano M,Lera D.Test functions with variable attraction regions for global optimization problems[J].Journal of Global Optimization,1998,13(2):207-223.
Gablonsky J M,Kelley C T.A locally-biased form of the DIRECT algorithm[J].Journal of Global Optimization,2001,21:27-37.
He J,Watson L T,et al.Dynamic data structures for a direct search algorithm[J].Computational Optimization and Applications,2002,23(1):5-25.
Hinton G E,Osindero S and Teh Y.A fast learning algorithm for deep belief nets[J].Neural Computation,2006,18:1527-1554.
Hinton,G E and Salakhutdinov R R.Reducing the dimensionality of data with neural networks[J].Science,2006,313(5786):504-507.
Holland J H.Adaption in nature and artificial systems.2nd ed.Cambrige,MA:MIT Press,1992.
Holmstrom K,Goran A O,Edvall M M.User's Guide for TOMLAB 7.Tomlab optimization.http://tomopt.com.
Huyer W,Neumaier A.Global optimization by multilevel coordinate search[J].Journal of Global Optimization,1999,14(4):331-355.
Jones D R,Perttunen C D,Stuckman B E.Lipschitzian optimization without the Lipschitz constant[J].Journal of Optimization Theory and Application,1993,79(1):157-181.
Jones D R.The DIRECT global optimization algorithm.In Floudas C.A.,Pardalos:The Encyclopedia of Optimization.Kluwer Academic,2001.
Kennedy J,Eberhart R C.Particle swarm optimization,In:proceeding of the IEEE iternational conference on neural networks IV,Piscataway:IEEE,1995,1942-1948,
LeCun Y,Bengio Y,Hinton G E.Deep Learning.Nature,2015,521:436-444.
Liu Q F.Linear scaling and the DIRECT algorithm[J].Journal of Global Optimization,2013,56(3):1233-1245.
Liu Q F.Order-2 stability analysis of particle swarm optimization[J].Evolutionary Computation,2015,23:187-216.
Liu Q F,Cheng W Y.A modified DIRECT algorithm with bilevel partition[J].Journal of Global Optimization,2014,60(3):483-499.
Liu Q F,Zeng J P.Global optimization by multilevel partition[J].Journal of Global Optimization,2015,61(1):47-69.
Liu Q F,Zeng J P,Yang G.MrDIRECT:A multilevel robust DIRECT algorithm for global optimization problems[J].Journal of Global Optimization,2015,62(2):205-227.
Liu Q F,Yang G,Zhang Z Z,Zeng J P.Improving the convergence rate of the DIRECT global optimization algorithm[J].Journal of Global Optimization,2017,67(4):851-872.
Liuzzi G,Lucidi S,Piccialli V.A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems[J].Computational Optimization and Applications,2010,45(2):353-375.
Liuzzi G,Lucidi S,Piccialli V.A partion-based global optimization algorithm[J].Journal of Global Optimization,2010,48:113-128.
Liuzzi G,Lucidi S,Piccialli V.Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization.Computational Optimization and Applications.DOI:10.1007/s10589-015-9741-9,2015.
Ljunberg K,Holmgren S.Simultaneous search for multiple QTL using the global optimization algorithm DIRECT[J].Bioinformatics,2004,20(12):1887-1895.
Moré J,Wild S.Benchmarking Derivative-Free Optimization Algorithms[J].SIAM Journal on Optimization,2009,20(1):172-191.
Pardalos P M,Schoen F.Recent advances and trends in global optimization:deterministic and stochastic methods.In:Proceedings of the Sixth International Conference on Foundations of Computer-Aided Process Design,DSI 1-2004,2004,119-131.
Paulavi?ius R,Sergeyev Y D,Kvasov D E and?ilinskas J.Globally-biased Disimpl algorithm for expensive global optimization[J].Journal of Global Optimization,2014,59:545-567.
Rios L M,Sahinidis N V.Derivative-free optimization:a review of algorithms and comparison of software implementations[J].Jouranl of Global Optimization,2013,56(3):1247-1293.
Sergeyev Y D,Kvasov D E.Global search based on efficient diagonal partitions and a set of Lipschitz constants[J].SIAM Journal on Optimization,2006,16(3):910-937.
Tawarmalani M,Sahinidis N V.A polyhedral branch-and-cut approach to global optimization[J].Mathematical Programming,2005,103(2):225-249.
ilinskas A.On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions[J].Applied Mathematics and Computation,2012,218(16):8131-8136.