Evolution Strategies An Example Evolution Strategy Procedure ES

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; }

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.

Evolution Strategies: The Simple (1+1)-ES • The simple (1+1)-ES has the following attributes: – Individuals are represented as follows: • – <xi, 0, xi, 1, …, xi, n-1, i>, where n is the number of variables Offspring are created as follows: +i, j = k, j * exp( 0 * N(0, 1)); x +i, j = xk, 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.

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

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?

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/back 91 survey. 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.

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

Evolution Strategies: The ( + )-ES • In the ( + )-ES, an individual, i, is represented as follows: • <xi, 0, xi, 1, …, xi, n-1, i, 0, i, 1, …, i, n-1>, 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 = xk, j + +i, j. N +i, j(0, 1); - Where j represents the jth variable, - ’ 1/sqrt(2 n) /* Global Learning Rate */ - 1/sqrt(2*sqrt(n)) /* Individual Learning Rate */ • The 1/5 th success rule is used.

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(<xk, yk, k, x, k, y>, <x +i, y +i, x, +i, y>): +i, x = k, x * exp( ’ * N(0, 1) + * N +i(0, 1)); x +i = xi + +i, x N +i, x(0, 1); +i, y = k, y * exp( ’ * N(0, 1) + * N +i(0, 1)); y +i = yi + +i, y N +i, y(0, 1); fit +i = Evaluate(<x +i, y +i>); } P(t+1) = Best of the + individuals; t = t + 1; } }

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(<xk, yk, k, x, k, y>, <x +i, y +i, x, +i, y>): +i, x = k, x * exp( ’ * N(0, 1) + * N +i(0, 1)); x +i = xi + +i, x N +i, x(0, 1); +i, y = k, y * exp( ’ * N(0, 1) + * N +i(0, 1)); y +i = yi + +i, y N +i, y(0, 1); fit +i = Evaluate(<x +i, y +i>); } P(t+1) = Best of the offspring; t = t + 1; } }