1. Evolution Strategies An Example Evolution Strategy Procedure ES{ t = 0; Initialize P(t); Evaluate P(t); While (Not Done) { Parents(t) = Select_Parents(P(t)); Offspring(t) = Procreate(Parents(t)); Evaluate(Offspring(t)); P(t+1)= Select_Survivors(P(t),Offspring(t)); t = t + 1; }

2. Evolution Strategies There are basically 4 types of ESs –The Simple (1+1)-ES –The (  +1)-ES (The first multimembered ES) –The (  + )-ES, and –The ( , )-ES.

3. Evolution Strategies: The Simple (1+1)-ES The simple (1+1)-ES has the following attributes: –Individuals are represented as follows:, where n is the number of variables –Offspring are created as follows:   +i,j =  k,j * exp(  0 * N(0,1)); x  +i,j = x k,j +   +i,j N  +i,j (0,1); Where j represents the jth variable. And where  0  1/sqrt(n)(Global Learning Rate) –Uses the 1/5 Success Rule to Adapt the Step Size: If more than 1/5 th of the mutations cause an improvement (in the objective function) then multiply  by 1.2, If less than 1/5 th of the mutations cause an improvement, then multiply  by 0.8.

4. Evolution Strategies: The Simple (1+1)-ES Procedure simpleES{ t = 0; Initialize P(t); /*  = 1, = 1 */ Evaluate P(t); while (t <= (4000-  )/ ){ for (i=0; i<1; i++){ Create_Offspring(, ):   +i =  i * exp(  0 * N(0,1)); x  +i = x i +   +i N  +i,x (0,1); y  +i = y i +   +i N  +i,y (0,1); fit  +i = Evaluate( ); } P(t+1) = Better of 2 individuals; t = t + 1; }

5. Evolution Strategies: The Simple (1+1)-ES How is a simple (1+1)-ES similar to a (1+1)-Standard EP? In what ways are the two different?

6. Evolution Strategies: The (  +1)-ES Since the (  +1)-ES is multi-membered, crossover can be used. According to, Bäck, T., Hoffmeister, F, and Schwefel, H.-P. (1991). “A Survey of Evolution Strategies”, The Proceedings of the 4 th International Conference on Genetic Algorithms, R. K. Belew and L. B. Booker Eds., pp. 2-9, Morgan Kaufmann. [can be found at: http://citeseer.nj.nec.com/back91survey.html] http://citeseer.nj.nec.com/back91survey.html –Uniform Crossover (also referred to a discrete recombination) can be used on the variable values as well as the strategy parameter. Adaptation of the step-size is not used in the (  +1)-ES.

7. Evolution Strategies: The (  +1)-ES Procedure (  +1)-ES { t = 0; Initialize P(t); /* of  individuals */ Evaluate P(t); while (t <= (4000-  )){ Create_Offspring(, ):   +i =  i * exp(  0 * N(0,1)); x  +i = x i +   +i N  +i,x (0,1); y  +i = y i +   +i N  +i,y (0,1); fit  +i = Evaluate( ); P(t+1) = Best  of the  +1 individuals; t = t + 1; }

8. Evolution Strategies: The (  + )-ES In the (  + )-ES, an individual, i, is represented as follows:, where n is the number of variables Offspring are created by as follows: –   +i,j =  k,j * exp(  ’ * N(0,1) +  * N  +i (0,1)); x  +i,j = x k,j +   +i,j N  +i,j (0,1); -Where j represents the jth variable, -  ’  1/sqrt(2n) /* Global Learning Rate */ -   1/sqrt(2*sqrt(n))/* Individual Learning Rate */ The 1/5 th success rule is used.

9. Evolution Strategies: The (  + )-ES Procedure (  + )-ES { t = 0; Initialize P(t); /* of  individuals */ Evaluate P(t); while (t <= (4000-  )/ ){ for (i=0; i< ; i++){ Create_Offspring(, ):   +i,x =  k,x * exp(  ’ * N(0,1) +  * N  +i (0,1)); x  +i = x i +   +i,x N  +i,x (0,1);   +i,y =  k,y * exp(  ’ * N(0,1) +  * N  +i (0,1)); y  +i = y i +   +i,y N  +i,y (0,1); fit  +i = Evaluate( ); } P(t+1) = Best  of the  + individuals; t = t + 1; }

10. Evolution Strategies: The ( , )-ES Procedure ( , )-ES { t = 0; Initialize P(t); /* of  individuals */ Evaluate P(t); while (t <= (4000-  )/ ){ for (i=0; i< ; i++){ Create_Offspring(, ):   +i,x =  k,x * exp(  ’ * N(0,1) +  * N  +i (0,1)); x  +i = x i +   +i,x N  +i,x (0,1);   +i,y =  k,y * exp(  ’ * N(0,1) +  * N  +i (0,1)); y  +i = y i +   +i,y N  +i,y (0,1); fit  +i = Evaluate( ); } P(t+1) = Best  of the offspring; t = t + 1; }