8 A Review of Major Application Areas of Differential Evolution <s> Genetically Programmed Differential Evolution Mutation Operators <s> Particle Swarm Optimisers (PSOs) search using a set of interacting particles flying over the fitness landscape. These are typically controlled by forces that encourage each particle to fly back both towards the best point sampled by it and towards the swarm's best. Here we explore the possibility of evolving optimal force generating equations to control the particles in a PSO using genetic programming. <s> BIB001 </s> 8 A Review of Major Application Areas of Differential Evolution <s> Genetically Programmed Differential Evolution Mutation Operators <s> The hybridization and combination of different Evolutionary Algorithms to improve the quality of the solutions and to accelerate execution is a common research practice. In this paper, we utilize Genetic Programming to evolve novel Differential Evolution operators. The genetic evolution resulted in parameter free Differential Evolution operators. Our experimental results indicate that the performance of the genetically programmed operators is comparable and in some cases is considerably better than the already existing human designed ones. <s> BIB002
As previously mentioned, in this section we investigate the discover of new efficient DE mutation operators using GP BIB002 . This is possible since, mutation operators are simply the composition of elementary functions such as addition, subtraction, and multiplication, operating on the vectors that represent individuals of the DE population. To this end, the terminal set used for GP, included two numerical constants, the vector of the best so far DE individual, x best g , three vectors of different randomly selected DE individuals, and the fixed mutation constant μ employed by the DE mutation operators. In detail, the terminal set used in this study was, g }. The function set was F = {+, −, , }, where and are defined as follows: where the vectors x, y ∈ R n , with x = (x 1 , x 2 , . . . , x n ) and y = (y 1 , y 2 , . . . , y n ). Note that the operator utilizes a protected division; if the absolute value of the denominator is less 0.0001, then returns 1. The presentation of the problem and the fitness function typically define the space of candidate solutions for each particular problem. At present, more than one performance measure are applicable. One approach is to use the distance of the discovered minimizer from the global one to measure the operator's performance BIB001 . However, in many real life applications the location of the global minimizer is unknown. Conversely, the value of the global minimum could be known (for example when minimizing the sum of squares, a chemical or physical process, etc.). In this study, we defined a fitness function, suitable for general optimization tasks, which utilizes three benchmark optimization problems discussed in previous Section: the Shekel's Foxholes, the Corana Parabola and the Levy No. 5 test problems. More specifically, the performance of each operator was measured through the sum of the generations required to locate the global minimum on each benchmark function, plus the minimum function values that were discovered. It is known that the performance of the DE algorithm (like the performance of every other EA) can vary with the initial random individuals. To reduce the effect of the stochastic nature of DE, 10 independent evaluations were performed, and the final fitness was averaged. If the global minimum was not found after 100 generations DE terminated. Using this fitness function, we strain GP evolution towards obtaining DE operators capable of locating the global optimum, within a minimum number of generations.