État de l'art des méthodes d'« optimisation globale »

Berthiau, Gérard; Siarry, Patrick

RAIRO - Operations Research - Recherche Opérationnelle, Tome 35 (2001) no. 3, pp. 329-365.

We present a review of the main “global optimization” methods. The paper comprises one introduction and two parts. In the introduction, we recall some generalities about non linear constraint-less optimization and we list some classifications which have been proposed for the global optimization methods. We then describe, in the first part, various “classical” global optimization methods, most of which available long before the appearance of Simulated Annealing (a key event in this field). There exists plenty of papers and books dealing with these methods, and studying in particular their convergence properties. The second part of the paper is devoted to more recent or atypical methods, mostly issued from combinatorial optimization. The three main methods are “metaheuristics”: Simulated Annealing (and derived techniques), Tabu Search and Genetic Algorithms; we also describe three other less known methods. For these methods, theoretical studies of convergence are less abundant in the literature, and the use of convergence results is by far more limited in practice. However, the fitting of some of these techniques to continuous variables problems gave very promising results; that question is not discussed in detail in the paper, but useful references allowing to deepen the subject are given.

[1] E.H.L. Aarts et P.J.M. Van Laarhoven, Simulated annealing : Theory and applications. D. Reidel Publishing Company (1987). | MR | Zbl

[2] R.S. Anderssen, Global optimization, édité par R.S. Anderssen, L.S. Jennings et D.M. Ryan. Optimization, Univ. of Queensland Press, St Lucia (1972) 28-48. | MR | Zbl

[3] J.P. Barthélémy, G. Cohen et A. Lobstein, Complexité algorithmique et problèmes de communication. Masson, Collection CNET-ENST (1992). | MR | Zbl

[4] R. Battiti et G. Tecchiolli, The continuous reactive tabu search : Blending Combinatorial Optimization and Stochastic Search for Global Optimization. Ann. Oper. Res. 63 (1996) 53-188. | Zbl

[5] R.W. Becker et G.V. Lago, A global optimization algorithm, dans Proc. of the 8th Allerton Conference on Circuits and Systems Theory. Montecillo, Illinois (1970) 3-12.

[6] M. Bertocchi et C.D. Odoardo, A stochastic algorithm for global optimization based on threshold accepting technique, dans 11th European Congress on Operational Research EURO XI. Aachen, Germany (1991).

[7] G. Berthiau, La méthode du recuit simulé pour la conception des circuits électroniques : adaptation et comparaison avec d'autres méthodes d'optimisation. Thèse de Doctorat de l'École Centrale de Paris (1994).

[8] I.O. Bohachevsky, M.E. Johnson et M.L. Stein, Generalized Simulated Annealing for function optimization. Technometrics 28 (1986) 209-217. | Zbl

[9] F.H. Branin et S.K. Hoo, A method for finding multiple extrema of a function of $n$ variables, édité par F.A. Lootsma, Numerical methods of nonlinear optimization. Academic Press, London (1972). | MR | Zbl

[10] S.H. Brooks, A discussion of random methods for seeking maxima. Oper. Res. 6 (1958) 244-251.

[11] D.G. Brooks et W.A. Verdini, Computational experience with Generalized Simulated Annealing over continuous variables. Amer. J. Math. Management Sci. 8 (1988) 425-449. | MR | Zbl

[12] F. Catthoor, H. De Man et J. Vandewalle, SAMURAI : A general and efficient simulated annealing schedule with fully adaptive annealing parameters. Integration, The VLSI Journal 6 (1988) 147-178.

[13] V. Cerny, Minimization of continuous functions by simulated annealing, Internal Documentation HU-TFT-84-51. Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20c, SF-00170, Helsinki 17, Finland (1984).

[14] V. Cerny, Thermodynamical approach to the traveling salesman problem : An efficient simulation algorithm. J. Optim. Theory Appl. 45 (1985) 41-51. | MR | Zbl

[15] I. Charon et O. Hudry, Le bruitage : une méthode prometteuse d'optimisation combinatoire. ENST, Département d'Informatique, Rapport Interne Télécom Paris 92-D-005 (1992).

[16] Y. Cherruault et A. Guillez, Une méthode pour la recherche du minimum global d'une fonctionnelle. C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 175-178. | Zbl

[17] Y. Cherruault, Mathematical modelling in Biomedicine. D. Reidel Publishing Company (1986). | MR

[18] Y. Cherruault, A new method for global optimization (Alienor). Kybernetes 19 (1989) 19-32. | MR | Zbl

[19] A. Corana, M. Marchesi, C. Martini et S. Ridella, Minimizing multimodal functions of continuous variables with the “simulated annealing” algorithm. ACM Trans. Math. Software 13 (1987) 262-280. | Zbl

[20] P. Courrieu, Un algorithme de recherche distribuée pour l'optimisation difficile. Univ. de Provence, Centre de Recherche en Psychologie Cognitive, Rapport Interne TF-9101 (1991).

[21] M. Creutz, Microcanonical Monte-Carlo simulation. Phys. Rev. Lett. 50 (1983) 1411-1414. | MR

[22] D. Cvijovic et J. Klinowski, Taboo search. An approach to the Multiple Minima Problem. Science 667 (1995) 664-666. | MR

[23] A. Dekkers et E.H.L. Aarts, Global optimization and simulated annealing. Math. Programming 50 (1991) 367-393. | MR | Zbl

[24] L. Devroye, Progressive global random search of continuous functions. Math. Programming 15 (1978) 330-342. | MR | Zbl

[25] D. De Werra et A. Hertz, Tabu search techniques : A tutorial and an application to neural networks. OR Spektrum 11 (1989) 131-141. | MR | Zbl

[26] L.C.W. Dixon et G.P. Szegö, Towards global optimization. North Holland, Amsterdam (1975). | Zbl

[27] L.C.W. Dixon et G.P. Szegö, Towards global optimization 2. North Holland, Amsterdam (1978). | MR | Zbl

[28] G. Dueck et T. Scheuer, Threshold accepting. IBM Zentrum Heidelberg, Germany (1989).

[29] G. Dueck, New optimization heuristics, the great deluge and the record-to record travel. J. Comput. Phys. 104 (1993) 86-92. | Zbl

[30] S. Geman et C.R. Hwang, Diffusions for global optimization. SIAM J. Control Optim. 24 (1986) 1031-1043. | MR | Zbl

[31] M. Gendreau, A. Hertz et G. Laporte, A Tabu Search Algorithm for the Vehicle Routing Problem. Management Sci. 40 (1994) 1276-1290. | Zbl

[32] F. Glover, Future paths for integer programming and links to artificial intelligence. Comput. Oper. Res. 13 (1986) 533-549. | MR | Zbl

[33] F. Glover et H.J. Greenberg, New approaches for heuristic search : A bilateral linkage with artificial intelligence. Eur. J. Oper. Res. 39 (1989) 119-130. | MR | Zbl

[34] F. Glover, Tabu search fundamentals and uses, Working paper. Graduate School of Business, Box 419, University of Colorado, Boulder, CO (1995).

[35] F. Glover et M. Laguna, Tabu search. Kluwer Academic Publishers (1997). | MR | Zbl

[36] D.E. Goldberg, Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading (1989). | Zbl

[37] L. Hérault, Réseaux de neurones récursifs pour l'optimisation combinatoire ; Application à la théorie des graphes et à la vision par ordinateur, Thèse de Doctorat de l'Institut National Polytechnique de Grenoble. INPG, Grenoble (1989).

[38] J.H. Holland, Adaptation in natural and artificial systems. Univ. of Michigan Press, Ann Arbor (1975). | MR | Zbl

[39] R. Horst, P.M. Pardalos, Handbook of Global Optimization. Kluwer Academic Publishers (1995). | MR | Zbl

[40] N. Hu, Tabu Search method with random moves for globally optimal design. Int. J. Numer. Meth. Eng. 35 (1992) 1055-1070.

[41] R.B. Kearfott, Test results for an interval branch and bound algorithm for equality-constrained optimization, édité par C. Floudas et P.M. Pardalos, State of the Art in Global Optimization : Computational Methods and Applications. Kluwer, Dordrecht, Netherlands (1996) 181-200. | MR | Zbl

[42] R.B. Kearfott et V. Kreinovich, Applications of Interval Computations. Kluwer, Dordrecht, Netherlands, Applied Optimization (1996). | MR | Zbl

[43] S. Kirkpatrick, C.D. Gelatt et M.P. Vecchi, Optimization by simulated annealing, Research Report RC 9355. IBM, Yorktown Heights, NY (1982).

[44] S. Kirkpatrick, C.D. Gelatt et M.P. Vecchi, Optimization by simulated annealing. Science 220 (1983) 671-680. | MR

[45] A.V. Levy et S. Gomez, The tunneling algorithm for the global optimization problem of constrained functions, Technical Report 231. Univ. Nat. Auton. de Mexico (1980).

[46] A.V. Levy et S. Gomez, The tunneling method applied to global optimization, édité par P.T. Boggs, R.H. Byrd et R.B. Schnanel, Numerical Optimization 1984. SIAM Philadelphia (1984). | MR | Zbl

[47] A.V. Levy et A. Montalvo, The tunneling algorithm for the global minimization of functions. SIAM J. Sci. Stat. Comp. 6 (1985) 15-29. | MR | Zbl

[48] M. Lundy et A. Mees, Convergence of an annealing algorithm. Math. Programming 34 (1986) 111-124. | MR | Zbl

[49] N. Metropolis, A.R. Rosenbluth, M.N. Rosenbluth, A. Teller et E. Teller, Equation of state calculations by fast computing machines. J. Chem. Phys. 21 (1953).

[50] Z. Michalewicz, Genetic algorithms + Data structures = Evolution Programs. Springer (1996). | MR | Zbl

[51] M. Minoux, Programmation mathématique - Théorie et algorithmes. Édition Dunod (1983). | Zbl

[52] R.E. Moore, On computing the range of values of a rational function of $n$ variables over a bounded region. Computing 16 (1976) 1-15. | MR | Zbl

[53] R.E. Moore, Methods and applications of interval analysis. SIAM, Philadelphia (1979). | MR | Zbl

[54] I. Mrad, La méthode du recuit simulé pour la synthèse automatique d'un schéma électrique équivalent. Application à la modélisation de composant et à l'adaptation à large bande. Thèse de Doctorat de l'École Centrale de Paris (1997).

[55] J.A. Nelder et R. Mead, A simplex method for function minimization. Comput. J. 7 (1965) 308-313. | Zbl

[56] C. Poivey, Méthodes d'optimisation globales pour la C.A.O. de circuits intégrés. Interface avec le simulateur SPICE-PAC. Thèse de Doctorat de l'Université de Clermont-Ferrand (1988).

[57] C.R. Reeves, Modern Heuristic Techniques for Combinatorial Problems. Mc Graw-Hill, Advanced Topics in Comput. Sci. Ser. (1995).

[58] A.H.G. Rinnooy Kan et G.T. Timmer, Stochastic methods for global optimization. Amer. J. Math. Management Sci. 4 (1984) 7-40. | MR | Zbl

[59] A.H.G. Rinnooy Kan et G.T. Timmer, Global optimization, Report 8612/A. Erasmus Univ. Rotterdam (1986).

[60] A.H.G. Rinnooy Kan et G.T. Timmer, Stochastic global optimization methods. Part I : Clustering methods. Math. Programming 39 (1987) 27-56. | MR | Zbl

[61] A.H.G. Rinnooy Kan et G.T. Timmer, Stochastic global optimization methods. Part II : Multi-level methods. Math. Programming 39 (1987) 57-78. | MR | Zbl

[62] E. Rolland, A Tabu Search Method for Constrained Real-Number Search : Applications to Portfolio Selection, Working Paper. The A. Gary Anderson Graduate School of Management, University of California, Riverside (1996).

[63] E. Rolland et H. Johnson, Skewness and the Mean-Variance Frontier : A Tabu Search Approach, Working Paper. The A. Gary Anderson Graduate School of Management, University of California, Riverside (1996).

[64] G. Rudolph, Convergence analysis of canonical genetic algorithms. IEEE Trans. Neural Networks 5 (1994) 96-101.

[65] K. Schittkowski et W. Hock, Test examples for nonlinear programing codes. Springer-Verlag, Lecture Notes in Econom. and Math. Systems 187 (1981). | MR | Zbl

[66] K. Schittkowski et W. Hock, More test examples for nonlinear programing codes. Springer-Verlag, Lecture Notes in Econom. and Math. Systems 282 (1988). | MR | Zbl

[67] P. Siarry, La méthode du recuit simulé : application à la conception de circuits électroniques. Thèse de Doctorat de l'Université Pierre et Marie Curie, Paris 6 (1986).

[68] P. Siarry et G. Dreyfus, La méthode du recuit simulé : théorie et applications. Éditeur IDSET (1988).

[69] P. Siarry, La méthode du recuit simulé en électronique : adaptation et accélération. Comparaison avec d'autres méthodes d'optimisation. Application dans d'autres domaines, Rapport d'habilitation à diriger les recherches en sciences. Université de Paris Sud, Centre d'Orsay (1994).

[70] P. Siarry, G. Berthiau, F. Durbin et J. Haussy, Enhanced Simulated Annealing for globally minimizing functions of many-continuous variables. ACM Trans. Math. Software 23 (1997) 209-228. | MR | Zbl

[71] P. Siarry et G. Berthiau, Fitting of Tabu Search to optimize functions of continuous variables. Int. J. Numer. Methods Eng. 40 (1997) 2449-2457. | MR | Zbl

[72] E.G. Talbi, A taxonomy of hybrid meta-heuristics, Rapport AS-183 du Laboratoire d'Informatique Fondamentale de Lille. Université des Sciences et Technologies de Lille (1998).

[73] A. Törn, A search clustering approach to global optimization, édité par L.C.W. Dixon et G.P. Szegö. North Holland, Amsterdam, Towards Global Optimization 2 (1978). | Zbl

[74] A. Törn et A. Zilinskas, Global optimization, édité par G. Goos et J. Hartmanis. Springer Verlag, No. 350 (1989). | MR | Zbl

[75] D. Vanderbilt et S.G. Louïe, A Monte-Carlo simulated annealing approach to optimization over continuous variables. J. Comput. Phys. 56 (1984) 259-271. | MR | Zbl