Optimizing complex systems by means of adaptively controlled random search

OPTII,IIZING COMPLEX SYSTEillS BY :.lEANS OF ADAPTIVELY CONTROLLED RANDOM SEARCH Joachim Born 1) Summary. X general class of random search methods for optimizing complex systems is considered. The special aim of this paper is the presentation of a uniform procedure to control the step sizes during the search process. A so-called variance function approach is proposed. Keywords. Optimizing complex systems, random search, global optimization, adaptive control. The task of optimizing a complex system arises in many areas of science. Frequently, the following optimization problem will occur:

f : R"-rR'

inf c f(x) : xex s X = { xc Rn : alxeb 19

a,btR", aiCbi , I= l,...,n. The function f is called the objective function. X is the feasible set. The objective function has not to be given by an analytic formula, but a unique value f(x) for every parameter vector x is required. Generally, we can not suppose "good-natured" properties of the objective function, like convexity, differentiability or unimodality. It can be stated that random search techniques are convenient methods to approximate a solution of such a general optimization problem. The main advantage of random search methods is their potential ability for finding a global optimum in a multiextremal case. During the last years a number of papers has been published towards adaptively controlled random methods, e.g. /4/,/8/,/g/,/10/,/12/. In these papers different procedures to control the step sizes (variances) are regarded reflecting the important role of a problemoriented variance adaptation for the efficiency of the methods. In /6/ a uniform procedure is proposed to control the variances by where

and

means of so-called variance functions. In order to explain this approach to a general algorithm easily we consider the one-dimensional case without loss of generality, only.

1) Central Institute for Cybernetics and Information Processes of the Academy of Sciences of the G.D.R., DDR-7086 3erlin, Kurstr. 33

Then the basic algorithm is given by the following framework. start with a point

x't X and then use an iterative procedure -

until a termination criterion is fulfilled: Generate a point y by y = NCxk,s) and set xk+l'=

y , if

yex

and

f(y)
L xk , otherwise. Replace k by k+lland repeat the procedure.

(By N(e,s) a random variable is denoted, normally distributed with expachation e and variance 8,) At this the vartance 8 is generated randomly by means of a variance function. As a variance function any distribution function can be used, whose density function is u&modal, having a bounded spectrum and depending on a mode m and a dispersion d, only. The mode m is to control adaptively, intending an increase of the efficiency of the search. In /6/ two procedures are proposed to carry out such a control, analysing the improvements of the objective function on account of some Iterations. The dispersion d is always unchanged. This is to disturb an exclusively local-oriented adaptation of the search. Choosing a variance function and setting the parameters sl,su,m,d, ths mode control, it is possible to generate numerically different variants of the general algorithm. The global convergence almost everywhere of the general algorithm is proved for any continuous objective function, In the paper a proposal towards a computationally useful type of a variance function will be given. Some special variants of the resulting method will be presented with regard to their computational properties using test results of different objective funtions. Finally, the software package ASTUP will be briefly outlined which were developed as an interactive workstation for this class of methods, References. /?/ Archetti,F.;G.Cugiani: Numerical techniques for stochastic systems. North-Rollsnd, Amsterdam-Hew York-Oxford 1980. /2/ Baba,N.: Convergence of a random optimization method for constrained optfmization problems. J. of Opt. Theory and Appl. 33 ($98?>, 457-467. /3/ Bellmann,K.;J.Eorn: Numerical solution of adaptation problems by mean8 of an evolution etrategy. In: Podelling and optimization of complex systems, Proceedings of the IFIP-'EC7working conference,

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Novosibirsk, Lecture iJotesin Control and Information Sciences /4/

vol. 18, Springer, Berlin-Heidelberg-Xew York 1978, 157-167. Born,J.;K.Yellmann: ?Jumericaladaptation of parameters in simulation models using evolution strategies. In: h.ijellmann(ed.): 'ioleculargenetic information systems: modelling and simulation. Akademie-Verlag, :3erlin 1983, 291-320.

/5/

Born,J.: ASTOP-I1andbook.Atechnical report, 1984. /6/ Born,J.: Adaptively controlled random search- a variance function approach. Cyst. Anal. .Iodel.Simul. 2(1985) 2, 105-116. /7/ aixon,L.C.d.;G.P.Szegoe (eds.): Towards global optimization vol. 1(1975), vol. 2(1978). North-Holland, Amsterdam-Xew YorkOxford. /8/ ;.!arti,K.: Random search in optimization problems as a stochastic decision process (adaptive random search). .leth.of Oper. iies.

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(1980), 223-234. /9/ Price,;/.L.:Global optimization by controlled random search. Journal of Opt. Theory and Appl. 40(1983) 3, 333-349. /lo/ Jasfrigin,L.A.;K.K.5?ipa;G.S.Tarasenko:Adaptation of random search (in Russian). Zinatne, Xiga 1978. /ll/ Rechenberg,I.: Evolutionsstrategie: Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. Frommann-Holzboog, Stuttgart 1973. mit/12/ Schwefe1,H.P.: ?:umerischeOptimierung von Computer-:.;odellen tels der Evolutionsstrategie. Interdisciplinary systems research vol. 26. Birkhauser, Base1 and Stuttgart 1977. Engl. ed.: Numerical optimization of computer models. J. T/iley&Sons, Chichester-Bew York-Brisbane-Toronto 1981.

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