8 A Review of Major Application Areas of Differential Evolution <s> EndIf Step 7. EndDo <s> A simple optimization procedure for constraint-based problems is described which works with a simplified cost function or even without one. The simplification of the problem formulation makes this method particularly attractive. The new method lends itself to parallel computation and is well suited for constraint satisfaction, constrained optimization, and design centering problems. A further asset is its self-steering property which makes the new method easy to use. <s> BIB001 </s> 8 A Review of Major Application Areas of Differential Evolution <s> EndIf Step 7. EndDo <s> Oligonucleotide arrays can provide a broad picture of the state of the cell, by monitoring the expression level of thousands of genes at the same time. It is of interest to develop techniques for extracting useful information from the resulting data sets. Here we report the application of a two-way clustering method for analyzing a data set consisting of the expression patterns of different cell types. Gene expres- sion in 40 tumor and 22 normal colon tissue samples was analyzed with an Affymetrix oligonucleotide array comple- mentary to more than 6,500 human genes. An efficient two- way clustering algorithm was applied to both the genes and the tissues, revealing broad coherent patterns that suggest a high degree of organization underlying gene expression in these tissues. Coregulated families of genes clustered together, as demonstrated for the ribosomal proteins. Clustering also separated cancerous from noncancerous tissue and cell lines from in vivo tissues on the basis of subtle distributed patterns of genes even when expression of individual genes varied only slightly between the tissues. Two-way clustering thus may be of use both in classifying genes into functional groups and in classifying tissues based on gene expression. <s> BIB002 </s> 8 A Review of Major Application Areas of Differential Evolution <s> EndIf Step 7. EndDo <s> Previous studies have shown that differential evolution is an efficient, effective and robust evolutionary optimization method. However, the convergence rate of differential evolution in optimizing a computationally expensive objective function still does not meet all our requirements, and attempting to speed up DE is considered necessary. In this paper, a new local search operation, trigonometric mutation, is proposed and embedded into the differential evolution algorithm. This modification enables the algorithm to get a better trade-off between the convergence rate and the robustness. Thus it can be possible to increase the convergence velocity of the differential evolution algorithm and thereby obtain an acceptable solution with a lower number of objective function evaluations. Such an improvement can be advantageous in many real-world problems where the evaluation of a candidate solution is a computationally expensive operation and consequently finding the global optimum or a good sub-optimal solution with the original differential evolution algorithm is too time-consuming, or even impossible within the time available. In this article, the mechanism of the trigonometric mutation operation is presented and analyzed. The modified differential evolution algorithm is demonstrated in cases of two well-known test functions, and is further examined with two practical training problems of neural networks. The obtained numerical simulation results are providing empirical evidences on the efficiency and effectiveness of the proposed modified differential evolution algorithm. <s> BIB003
We now briefly review the two basic DE variation operators. The first DE operator we consider is mutation. Specifically, for each individual x  where x best g is the best member of the previous generation; μ > 0 is a real parameter, called mutation constant, which controls the amplification of the difference between two individuals so as to avoid the stagnation of the search process; and r 1 , r 2 , r 3 , r BIB002 , r 5 ∈ {1, 2, . . . , i − 1, i + 1, . . . , NP }, are random integers mutually different and not equal to the running index i. Trying to rationalize the above equations, we observe that Equation (2) is similar to the crossover operator used by some Genetic Algorithms and Equation (1) derives from it, when the best member of the previous generation is employed. Equations (3), (4) and (5) are modifications obtained by the combination of Equations (1) and  . It is clear that more such relations can be generated using the above ones as building blocks. For example, the recently proposed trigonometric mutation operator BIB003 performs mutations with probability τ μ according to the following equation:  and with probability (1−τ μ ) performs mutations according to Equation  , where τ μ is a user defined parameter. The values of p m , m = {1, 2, 3} and p are obtained through the following equations: For the rest of the chapter, we call DE 1 the differential evolution algorithm that uses Equation (1) as the mutation operator, DE 2 the algorithm that uses Equation  , and so on. The recombination operator is subsequently applied to further increase the diversity of the mutant individuals. To this end, the resulting individuals are combined with other predetermined individuals, called the target individuals. Specifically, for each component l (l = 1, 2, . . . , n) of the mutant individual v To prevent an individual from surviving indefinitely, we can employ the concept of aging BIB001 . To this end, each vector is randomly assigned a maximum age, i.e. an integer from the interval [α, β] , where α and β are the minimum and the maximum possible age, respectively. At each iteration, the age of each vector is increased by one, and if it exceeds its maximum age then the individual "dies". This individual is then replaced by another vector randomly chosen from the current population. Note that it is desirable the best individual of the population not to be eliminated.