A Combined Nelder - Mead Simplex and Geneti

Ni olas Durand Laboratoire d'Optimisation Globale Centre d'Etudes de la Navigation A erienne 7, av Edouard Belin 31055 Toulouse Cedex Fran e durand re her he.ena .fr tel: (33) 562 17 40 54 Jean-Mar Alliot Laboratoire d'Optimisation Globale Centre d'Etudes de la Navigation A erienne 7, av Edouard Belin 31055 Toulouse Cedex Fran e alliot re her he.ena .fr tel: (33) 562 17 41 24 Abstra t It is usually said that geneti algorithm should be used when nothing else works. In pra ti e, geneti algorithm are very often used for large sized global optimization problems, but are not very eÆ ient for lo al optimization problems. The Nelder-Mead simplex algorithm has some ommon hara teristi s with geneti algorithm, but it an only nd a lo al optimum lose to the starting point. In this arti le, a ombined NelderMead Simplex and Geneti algorithm is introdu ed and tested on lassi al test fun tions on whi h both geneti algorithm or loal optimization te hniques are not eÆ ient when separately used. Introdu tion Geneti algorithm are not very fast to solve lo al optimization problems but an be very powerful to nd a global optimum area. However, it is sometimes very diÆ ult to nd the minimum of a fun tion using a geneti algorithm be ause bad solutions an be very near to the global optimum so that when the geneti algorithm is unlu ky it may have some problems to nd and remain in good areas. Lo al optimization te hniques su h as the Nelder-Mead Simplex have some ommon hara teristi s with geneti algorithm as they do not use the su essive derivatives of the fun tion and deals with a population of points instead of a single point. Furthermore, they are quite eÆ ient to nd a lo al optimum very qui kly. In this arti le a te hnique that ombines the Nelder-Mead Simplex and Geneti algorithm is de ned and tested on lassi al test fun tions on whi h geneti algorithm or lo al optimization te hniques are not eÆ ient. -1 -0.5 0 0.5 1 1.5