An information global minimization algorithm using the local improvement technique

In this paper, the global optimization problem with a multiextremal objective function satisfying the Lipschitz condition over a hypercube is considered. An algorithm that belongs to the class of information methods introduced by R.G. Strongin is proposed. The knowledge of the Lipschitz constant is not supposed. The local tuning on the behavior of the objective function and a new technique, named the local improvement, are used in order to accelerate the search. Two methods are presented: the first one deals with the one-dimensional problems and the second with the multidimensional ones (by using Peano-type space-filling curves for reduction of the dimension of the problem). Convergence conditions for both algorithms are given. Numerical experiments executed on more than 600 functions show quite a promising performance of the new techniques.

[1]  Yaroslav D. Sergeyev,et al.  An Information Global Optimization Algorithm with Local Tuning , 1995, SIAM J. Optim..

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

[3]  Tibor Csendes,et al.  Developments in Global Optimization , 1997 .

[4]  Jon G. Rokne,et al.  New computer methods for global optimization , 1988 .

[5]  A. Zhigljavsky.  Stochastic Global Optimization , 2008, International Encyclopedia of Statistical Science.

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

[7]  S. A. Piyavskii.  An algorithm for finding the absolute extremum of a function , 1972 .

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

[9]  Yaroslav D. Sergeyev,et al.  Global one-dimensional optimization using smooth auxiliary functions , 1998, Math. Program..

[10]  Roman G. Strongin,et al.  The information approach to multiextremal optimization problems , 1989 .

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

[12]  Pierre Hansen,et al.  Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison , 1989, Math. Program..

[13]  Pasquale Daponte,et al.  Two methods for solving optimization problems arising in electronic measurements and electrical engineering , 1999, SIAM J. Optim..

[14]  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 .

[15]  Y. Evtushenko.  Numerical methods for finding global extrema (Case of a non-uniform mesh) , 1971 .

[16]  Yaroslav D. Sergeyev,et al.  Algorithm 829: Software for generation of classes of test functions with known local and global minima for global optimization , 2003, TOMS.

[17]  Harold J. Kushner,et al.  A New Method of Locating the Maximum Point of an Arbitrary Multipeak Curve in the Presence of Noise , 1964 .

[18]  Arthur R. Butz,et al.  Space Filling Curves and Mathematical Programming , 1968, Inf. Control..

[19]  Yaroslav D. Sergeyev,et al.  Global Minimization Algorithms for Holder Functions , 2002 .

[20]  Roman G. Strongin,et al.  Global optimization with non-convex constraints , 2000 .

[21]  Roman G. Strongin,et al.  Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves , 1992, J. Glob. Optim..

[22]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[23]  John J. Bartholdi,et al.  Spacefilling curves and the planar travelling salesman problem , 1989, JACM.