Evolutionary Programming and Evolution Strategies Similarities and Dierences y z Thomas Back Gunter Rudolph HansPaul Schwefel University of Dortmund Department of Computer Science Chair of Systems Analysis PO Box Dortmund Germany gorithms to realworld problems have clearly demon Abstract strated their capability to yield go o d approximate so Evolutionary Programming and Evolution Strategies lutions even in case of complicated multimo dal top o rather similar representatives of a class of probabilis logical surfaces of the tness landscap e for overviews tic optimization algorithms gleaned from the mo del of of applications the reader is referred to the conference organic evolution are discussed and compared to each pro ceedings or to the annotated other with resp ect to similarities and dierences of their bibliography basic comp onents as well as their p erformance in some exp erimental runs Theoretical results on global conver Until recentlydevelopment of these main streams was gence step size control for a strictly convex quadratic completely indep endentfromeachother Since function and an extension of the convergence rate the however contact b etween the GAcommunity and the ory for Evolution Strategies are presented and discussed EScommunity has b een established conrmed by col with resp ect to their implications on Evolutionary Pro lab orations and scientic exchange during regularly al gramming ternating conferences in the US International Con ference on Genetic Algorithms and their Applications ICGA since and in Europ e International Confer Intro duction ence on Parallel Problem Solving from Nature PPSN since Contact b etween the EPcommunity and Develop ed indep endently from each other three main the ES communit yhowever has b een established for the streams of socalled Evolutionary Algorithms ie algo rst time just in For algorithms b earing so much rithms based on the mo del of natural evolution as an similarities as ESs and EP do this is a surprising fact optimization pro cess can nowadays b e identied Evo elop ed by LJFogel lutionary Programming EP dev et al in the US Genetic Algorithms GAs devel Similar to a pap er comparing ES and GA ap op ed by J Holland also in the US and Evolution proaches the aim of this article is to giveanintro Strategies ESs develop ed in Germanyby I Rechen duction to ESs and EP and to lo ok for similarities and b erg and HPSchwefel dierences b etween b oth approaches A brief overview These algorithms are based on an arbitrarily initial of the historical development of ESs as well as an ex ized p opulation of searchpoints whichby means of ran planation of the basic algorithm are given in section domized pro cesses of selection mutation and some In section the basic EPalgorithm is presented Sec times recombination evolves towards b etter and b et tion then serves to discuss theoretical results from ESs ter regions in the search space Fitness of individ and their p ossible relations to the b ehaviour of an EP uals is measured by means of an ob jective function algorithm Section presents a practical comparison to b e optimized and several applications of these al of b oth algorithms using a few ob jective functions with dierent top ological shap es Finallyanoverview of simi baecklsinformatikunidortmundde y larities and dierences of b oth approaches is summarized rudolphlsinformatikunidortmundde z schwefellsinformatikunidortmundde in section f and f resp ectively This forms the basis of Rechen Evolution Strategies b ergs success rule Similar to EP ESs are also based on realvalued ob ject variables and normally distributed random mo dications with exp ectation zero According to Rechenb erg The ratio of successful mutations to al l muta rst exp erimental applications to parameter optimiza tions should be Ifitisgreater increase if tion p erformed during the middle of the sixties at the it is less decrease the standard deviation Technical University of Berlin dealt with hydro dynam ical problems like shap e optimization of a b ended pip e and a ashing nozzle The algorithm used was a sim For this algorithm Schwefel suggested to measure ple mutationselection scheme working on one individ the success probability p by observing this ratio during ual which created one ospring by means of mutation the search and to adjust t according to The b etter of parent and ospring is selected determin istically to survive to the next generation a selection mechanism whichcharacterizes this two membered or n t c if p ES Assuming inequality constraints g IR j t t c if p IRj fvg of the search space an ob jective n t if p function f M IR IR where the feasible region M is dened by the inequality constraints g a minimiza j n tion task and individual vectors xt IR where t n For the constant c he prop osed to use c Do denotes the generation counter a ES is dened ing so yields convergence rates of linear order in b oth y the following algorithm b mo del cases Algorithm ES The multimembered ES intro duces the concepts p op t ulation recombination and selfadaptation of strategy initalize P fxg parameters into the algorithm According to the selec such that j g x j P tion mechanism the ES and ES are dis while termination criterion not full led do tinguished the rst case indicating that parents cre mutate P tx txt z t ate ospring individuals by means of recombina with probability density tion and mutation The b est individuals out of parents p z t pz t exp i i and ospring are selected to form the next p opulation evaluate P tf xtfx t For a ES with the b est individuals are se select P t from P t lected from the ospring onlyEach individual is char if f x t f xt and j g x t j acterized not only byavector x of ob ject variables but then xt x t also by an additional vector of strategy variables The else xt xt latter may include up to n dierentvariances c ii i t t i fngaswellasupton n covari od ances c i fn g j fi ngofthe ij generalized ndimensional normal distribution having a To each comp onent of the vector xt the same stan probabilitydensity function dard deviation is applied during mutation The vari ation of ie the stepsize control of the algorithm is done according to a theoretically supp orted rule whichis s due to Rechenb erg For the ob jective functions f det A T exp z Az pz n a linear corridor of width b f x Fx c c x i fng b x b i and the spheremodel f Altogether up to w n n strategy parameters n can b e varied during the optimum searchby means of a X f x c c x x c c r i selectionmutationrecombination mechanism To assure i i p ositivedeniteness of the covariance matrix A the algorithm uses the equivalent rotation angles he calculated the optimal exp ected convergence rates j instead of the co ecients c The resulting from which the corresp onding optimal success probabil j ij algorithm reads as follows ities p and p can b e derived for opt opt the interval Besides completely missing recombi Algorithm ES ES nation the dierentvariants indicated are discrete t recombination intermediate recombination and initialize P fa a gI the global versions of the latter two resp ectively w where I IR Empirically discrete recombination on ob ject variables a x c c i j fng j i k i ij ji and intermediate recombination on strategy parameters evaluate P have b een observed to give b est results while termination criterion not full led do recombine a tr P t k fg k mutate a tma t Evolutionary Programming k k evaluate P tfa ta tg Following the description of an EP algorithm as given by ff x tfx tg Fogel and using the notational style from the previous select P t if ES section an EP algorithm is formulated as follows then sP t else sP t P t Algorithm EP t t od t initialize P fx x gI Then the mutation op erator must b e extended ac n where I IR cording to dropping time counters t evaluate P F x Gf x k k k while termination criterion not full led do I ma a x k k mutate x tmx t k fg k k p erforming comp onentwise op erations as follows evaluate P tfx tx tg fF x tF x tg exp exp i i i select P t sP t P t j j j t t x x z i i i od This waymutations of ob ject variables are correlated Besides a missing recombination op erator tness eval according to the values of the vector and provides a uation mutation and selection are dierent from cor scaling of the linear metrics Alterations and resp onding op erators in ESs Fitness values F x are i are again normally distributed with exp ectation zero p p obtained from ob jective function values by scaling them n and variance one and the constants p to p ositivevalues function G and p ossibly by imp osing n and are rather robust ex some random alteration For mutation the standard i ogenous parameters scaled by is a global fac deviation for each individuals mutation is calculated as tor identical for all i fng whereas is an i the square ro ot of a linear transformation of its own t individual factor sampled anew for all i fng ness value ie for mutation mxx i fng allowing of individual changes of mean step sizes i Concerning recombination dierent mechanisms can x z x i i i p b e used within ESs where in addition to the usual re Fx i i i combination of two parents global mechanisms allowfor taking into account up to all individuals of a p opulation Again the random variable z has probability