A modification of the DIRECT method for Lipschitz global optimization for a symmetric function

In this paper, we consider a global optimization problem for a symmetric Lipschitz continuous function. An efficient modification of the well-known DIRECT (DIviding RECTangles) method called SymDIRECT is proposed for solving this problem. The method is illustrated and tested on several standard test functions. The application of this method to solving complex center-based clustering problems for the data having only one feature is particularly presented.

[1]  Jianhong Wu,et al.  Data clustering - theory, algorithms, and applications , 2007 .

[2]  Yaroslav D. Sergeyev,et al.  Index branch-and-bound algorithm for Lipschitz univariate global optimization with multiextremal constraints , 2001, J. Glob. Optim..

[3]  Rudolf Scitovski,et al.  Solving the parameter identification problem of mathematical models using genetic algorithms , 2004, Appl. Math. Comput..

[4]  Christodoulos A. Floudas,et al.  A review of recent advances in global optimization , 2009, J. Glob. Optim..

[5]  J D Pinter,et al.  Global Optimization in Action—Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications , 2010 .

[6]  Yaroslav D. Sergeyev,et al.  Global Search Based on Efficient Diagonal Partitions and a Set of Lipschitz Constants , 2006, SIAM J. Optim..

[7]  Cem Iyigun,et al.  A generalized Weiszfeld method for the multi-facility location problem , 2010, Oper. Res. Lett..

[8]  B. P. Zhang,et al.  Estimation of the Lipschitz constant of a function , 1996, J. Glob. Optim..

[9]  Daniela di Serafino,et al.  A Modified DIviding RECTangles Algorithm for a Problem in Astrophysics , 2011, J. Optim. Theory Appl..

[10]  Yurij G. Evtushenko,et al.  Numerical Optimization Techniques , 1985 .

[11]  Daniel E. Finkel,et al.  Additive Scaling and the DIRECT Algorithm , 2006, J. Glob. Optim..

[12]  Linet Özdamar,et al.  TRIOPT: a triangulation-based partitioning algorithm for global optimization , 2005 .

[13]  Donald R. Jones,et al.  A Taxonomy of Global Optimization Methods Based on Response Surfaces , 2001, J. Glob. Optim..

[14]  J. Kogan.  Introduction to Clustering Large and High-Dimensional Data , 2007 .

[15]  C. D. Perttunen,et al.  Lipschitzian optimization without the Lipschitz constant , 1993 .

[16]  Yaroslav D. Sergeyev,et al.  A univariate global search working with a set of Lipschitz constants for the first derivative , 2009, Optim. Lett..

[17]  Tamara G. Kolda,et al.  Asynchronous parallel hybrid optimization combining DIRECT and GSS , 2010, Optim. Methods Softw..

[18]  Anne Auger,et al.  Theory of Randomized Search Heuristics , 2012, Algorithmica.

[19]  Eldon Hansen,et al.  Global optimization using interval analysis , 1992, Pure and applied mathematics.

[20]  Xin-She Yang,et al.  Firefly Algorithms for Multimodal Optimization , 2009, SAGA.

[21]  Lakhdar Chiter.  DIRECT algorithm: A new definition of potentially optimal hyperrectangles , 2006, Appl. Math. Comput..

[22]  Friedrich Leisch,et al.  A toolbox for K-centroids cluster analysis , 2006 .

[23]  Anne Auger,et al.  Theory of Randomized Search Heuristics: Foundations and Recent Developments , 2011, Theory of Randomized Search Heuristics.

[24]  Stefano Lucidi,et al.  A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems , 2010, Comput. Optim. Appl..

[25]  Yaroslav D. Sergeyev,et al.  Lipschitz gradients for global optimization in a one-point-based partitioning scheme , 2012, J. Comput. Appl. Math..

[26]  I. Alolyan.  A new exclusion test for finding the global minimum , 2007 .

[27]  S. Zlobec.  The fundamental theorem of calculus for Lipschitz functions , 2008 .

[28]  Lakhdar Chiter.  A new sampling method in the DIRECT algorithm , 2006, Appl. Math. Comput..

[29]  A. Neumaier.  Complete search in continuous global optimization and constraint satisfaction , 2004, Acta Numerica.

[30]  A. A. Zhigli︠a︡vskiĭ,et al.  Stochastic Global Optimization , 2007 .

[31]  Y. D. Sergeyev,et al.  Global Optimization with Non-Convex Constraints - Sequential and Parallel Algorithms (Nonconvex Optimization and its Applications Volume 45) (Nonconvex Optimization and Its Applications) , 2000 .

[32]  Marco Gaviano,et al.  A global minimization algorithm for Lipschitz functions , 2007, Optim. Lett..

[33]  Marc Teboulle,et al.  A Unified Continuous Optimization Framework for Center-Based Clustering Methods , 2007, J. Mach. Learn. Res..

[34]  Daniel Scholz,et al.  The big cube small cube solution method for multidimensional facility location problems , 2010, Comput. Oper. Res..

[35]  D. Finkel,et al.  Direct optimization algorithm user guide , 2003 .

[36]  Ying Zhang,et al.  A filled function method applied to nonsmooth constrained global optimization , 2009, J. Comput. Appl. Math..

[37]  Lonnie Hamm,et al.  GLOBAL OPTIMIZATION METHODS , 2002 .

[38]  Arnold Neumaier,et al.  Global Optimization by Multilevel Coordinate Search , 1999, J. Glob. Optim..

[39]  Yaroslav D. Sergeyev,et al.  Lipschitz Global Optimization , 2011 .

[40]  Y. Sergeyev,et al.  Univariate geometric Lipschitz global optimization algorithms , 2012 .

[41]  Zeev Volkovich,et al.  Building initial partitions through sampling techniques , 2007, Eur. J. Oper. Res..

[42]  C. Iyigun.  Probabilistic Distance Clustering , 2011 .

[43]  H. Spath,et al.  Cluster-Formation und Analyse. , 1984 .

[44]  Sanjo Zlobec.  Equivalent formulations of the gradient , 2011, J. Glob. Optim..

[45]  Jonas Mockus,et al.  On the Pareto Optimality in the Context of Lipschitzian Optimization , 2011, Informatica.

[46]  Robert J. Vanderbei,et al.  Extension of Piyavskii's Algorithm to Continuous Global Optimization , 1999, J. Glob. Optim..

[47]  B. Shubert.  A Sequential Method Seeking the Global Maximum of a Function , 1972 .

[48]  C. T. Kelley,et al.  Modifications of the direct algorithm , 2001 .

[49]  P. Pardalos,et al.  Handbook of global optimization , 1995 .

[50]  Tamara G. Kolda,et al.  Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods , 2003, SIAM Rev..

[51]  C. T. Kelley,et al.  A Locally-Biased form of the DIRECT Algorithm , 2001, J. Glob. Optim..

[52]  Kristian Sabo,et al.  One-dimensional center-based l1-clustering method , 2013, Optim. Lett..