A hybrid algorithm for identifying global and local minima when optimizing functions with many minima

The problem of determining most global minima including some of the local ones for unconstrained non-convex functions is investigated. This is tackled using a hybrid approach that combines simulated annealing, tabu search and a descent method. This approach has the advantage of not requiring differentiability of the function. The special feature of this approach is that it produces not only all the global minima with a high frequency but also those good local minima which may be of relevance to the user. This is the first time such a view is put forward, especially in the context of functions of continuous variables. This method is tested on standard test functions used in the literature, and encouraging results are obtained.

[1]  George A. Marcoulides,et al.  Modern methods for business research , 1998 .

[2]  Patrick Siarry,et al.  Tabu Search applied to global optimization , 2000, Eur. J. Oper. Res..

[3]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[4]  G. T. Timmer,et al.  Stochastic global optimization methods part II: Multi level methods , 1987, Math. Program..

[5]  David Connolly,et al.  General Purpose Simulated Annealing , 1992 .

[6]  Patrick Siarry,et al.  Enhanced simulated annealing for globally minimizing functions of many-continuous variables , 1997, TOMS.

[7]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[8]  K. Al-Sultan,et al.  A tabu search Hooke and Jeeves algorithm for unconstrained optimization , 1997 .

[9]  Roberto Battiti,et al.  The continuous reactive tabu search: Blending combinatorial optimization and stochastic search for global optimization , 1996, Ann. Oper. Res..

[10]  Wesley E. Snyder,et al.  Optimization of functions with many minima , 1991, IEEE Trans. Syst. Man Cybern..

[11]  C. Reeves.  Modern heuristic techniques for combinatorial problems , 1993 .

[12]  Nicos Christofides,et al.  Capacitated clustering problems by hybrid simulated annealing and tabu search , 1994 .

[13]  Emile H. L. Aarts,et al.  Global optimization and simulated annealing , 1991, Math. Program..

[14]  B. Golden,et al.  Using simulated annealing to solve routing and location problems , 1986 .

[15]  Ehl Emile Aarts,et al.  Statistical cooling : a general approach to combinatorial optimization problems , 1985 .

[16]  P. Siarry,et al.  FITTING OF TABU SEARCH TO OPTIMIZE FUNCTIONS OF CONTINUOUS VARIABLES , 1997 .

[17]  Jacek Klinowski,et al.  Taboo Search: An Approach to the Multiple Minima Problem , 1995, Science.

[18]  Said Salhi,et al.  Heuristic Search Methods , 1998 .

[19]  Patrick Siarry,et al.  A Continuous Genetic Algorithm Designed for the Global Optimization of Multimodal Functions , 2000, J. Heuristics.

[20]  H Burkhardt,et al.  Principles and problems of phase-shift analysis , 1975 .

[21]  H. Zimmermann.  Towards global optimization 2: L.C.W. DIXON and G.P. SZEGÖ (eds.) North-Holland, Amsterdam, 1978, viii + 364 pages, US $ 44.50, Dfl. 100,-. , 1979 .

[22]  Silvano Martello,et al.  Meta-Heuristics: Advances and Trends in Local Search Paradigms for Optimization , 2012 .

[23]  F. Aluffi-Pentini,et al.  Global optimization and stochastic differential equations , 1985 .

[24]  Saïd Salhi,et al.  A Tabu Search Approach for the Resource Constrained Project Scheduling Problem , 1998, J. Heuristics.