IGA for Solving Subproblems

Intelligent Crossover

The intelligent crossover plays an important role in IGA. IGA solves an individual sub-problem with N genes having 2(N + 1) parameters to be optimized. The intelligent crossover uses a divide-and-conquer approach, which consists of adaptively dividing two parents into α pairs of parameter groups, economically identifying the potentially better one of two groups of each pair, and systematically obtaining a potentially good approxi-mation to the best one of all 2^α combinations using at most 2α fitness evaluations. Like traditional GAs, two parents P₁and P₂produce two children C₁and C₂using one crossover operation. The intelligent crossover determines the recombination of P₁ and P₂ for ef-ficiently generating good children. Let the set of parameters in the i-th subproblem be {α_i, g_i1, . . . , g_iN, β_i, h_i1, . . . , h_iN}. We divided the two sets IN C = {α_i, g_i1, . . . , g_iN} and DEC = {β_i, h_i1, . . . , h_iN} which control the gene expression level increasing or decreasing into dα/2e and bα/2c groups, respectively. To make a sufficient use of all columns in OAs, α is usually set to 2^ω− 1 where ω is an integer. In this study, we used α = 7 for problems with N ≤ 30. The value of α would properly increase when N increases. The discussion between α and the number of parameters to be optimized can be referred to [38].

Because the parameters belonging to the same one of two sets IN C and DEC have strong interactions, we don’t use the conventional encoding scheme of GA that all

pa-rameters are encoded into a chromosome in a fixed order. Instead, all papa-rameters are represented using real values with no order. For each time using an intelligent crossover operation, IN C and DEC are randomly divided into dα/2e and bα/2c groups with a variable size for each group. The parameters of two parents are grouped using the same division operation. Each group is treated as a factor. The α factors are randomly num-bered in using OED. The numbering order does not affect the effectiveness of intelligent crossover because of the property of OA [38]. Note that there is no fixed genotype of S-system parameters used. The following steps describe how to use OED with α factors to achieve the intelligent crossover of IGA for a fitness function y.

Step 1: The two sets IN C = {αi, gi1, . . . , giN} and DEC = {βi, hi1, . . . , hiN} of S-system parameters are randomly divided into dα/2e and bα/2c groups (fac-tors), respectively.

Step 2: Use a two-level OA L_α+1(2^α) with α + 1 rows and α columns.

Step 3: Let levels 1 and 2 of factor d represent the d-th groups coming from parents P₁ and P₂, respectively.

Step 4: Evaluate the fitness values yt for experiment t where t = 2, . . . , α + 1. The value y₁ is the fitness value of P₁.

Step 5: Compute the main effect Sdk where d = 1, . . . , α and k = 1, 2.

Step 6: Determine the better one of two levels of each factor according to the main effect.

Step 7: The chromosome of C₁ is formed using the combination of the better groups from the derived corresponding parents.

Step 8: The chromosome of C₂ is formed similarly as C₁, except that the factor with the smallest main effect difference adopts the other level.

Step 9: The best two individuals among P₁, P₂, C₁, C₂, and α combinations of OA are used as the final children C1 and C2 for elitist strategy.

One intelligent crossover operation takes α + 2 fitness evaluations to explore the search space of 2^α combinations. Generally, C1 is a potentially good approximation to the best one of 2^α combinations.

Illustrative Example of Intelligent Crossover

Tables 5.1 and 5.2 show an illustrative example of using intelligent crossover with OED in solving the first subproblem of inferring an S-system model with N = 5. The details of the test problem are given in Section 5.3.1. We used an OA L8(2⁷) for α = 7. The two sets of S-system parameters IN C = {α₁, g₁₁, . . . , g₁₅} and DEC = {β₁, h₁₁, . . . , h₁₅} are randomly divided and assigned to four and three groups (factors) respectively as follows: V₁ = {h₁₃, h₁₅}, V₂ = {g₁₄}, V₃ = {g₁₂, g₁₃}, V₄ = {α₁, g₁₅}, V₅ = {h₁₁, h₁₂}, V₆ = {β1, h₁₄}, and V₇ = {g₁₁}. The parameter values of parents are given in Table 5.2. Table 5.1 shows all results of intelligent crossover using OED. First, we evaluate the response variable y_t of the combination t, where t = 1, 2, . . . , 8. Second, we compute the main effect Sdk where d = 1, 2, . . . , 7 and k = 1, 2. For example, S22= y3+ y4+ y7+ y8 = 147.65. Third, the better level of each factor based on the main effect is determined. For example, the better level of factor 1 is level 2 since S₁₂(153.97) < S₁₁(157.50). Finally, the better levels of factors (V₁, V₂, V₃, V₄, V₅, V₆, V₇) are (2, 2, 1, 1, 1, 2, 2) and then y = 30.22 can be obtained from the reasoned combination. This reasoned combination is used to form the child C₁ of the crossover operation. The least effective factor is d = 5 with M ED₅ = 2.06 which is the smallest one, so the second child C₂ is formed similarly as C1 except V5, which adopts level 2. Note that the ranks of C1 and C2 are 2 and 4 respectively among 128 combinations of a complete factorial experiment. It reveals that the reasoning operation of intelligent crossover for generating children is efficient.

The Used Intelligent Genetic Algorithm

IGA is used to solve the N individual subproblems with the fitness function Eq. 5.4. The gene expression level of X_cal,i,t is numerically calculated using Eq. 2.2 rather than Eq. 5.3 due to the following reasons:

1) According to the simulation using IGA, the method using Eq. 2.2 is simple and fast, and its solution is accurate enough in terms of fitness value from noise-free gene expression profiles.

2) We would further refine the combined solutions of the N subproblems from the aspect of global optimization using OSA for handling noisy gene expression profiles.

Table5.1.AnIllustrativeExampleofIntelligentCrossoverUsingOAL8(27 ). Factord CombinationtV1V2V3V4V5V6V7ytRank 1111111140.1281/128 2111222239.7072/128 3122112236.3423/128 4122221141.3597/128 5212121241.40102/128 6212212142.60125/128 7221122134.3213/128 8221211235.6518/128 Sd1157.60163.81149.78152.17154.70158.51158.39 Sd2153.97147.65161.68159.30156.76152.95153.08 MEDd3.5416.1611.907.132.065.565.31 Child1(C1)221112230.222/128 Child2(C2)221122230.804/128

Table5.2.TheContentsofParentsandChildren. α1g11g12g13g14g15β1h11h12h13h14h15y P12.982.080.98-1.172.130.002.712.040.101.892.33-0.0640.12 P212.680.58-0.240.790.330.8210.690.742.313.001.33-1.3332.68 C112.680.58-0.240.790.330.822.712.040.101.892.33-0.0630.22 C212.680.58-0.240.790.330.822.710.742.311.892.33-0.0630.80

3) The estimation method for ˆX_j in Eq. 5.3 using a cooperative coevolutionary algo-rithm on a PC cluster [31] is not suitable for the IGA-based method because that IGA solves each subproblem independently on a single-processor PC. Furthermore, the method using estimation of ˆX_j only slightly enhanced the probability of finding the correct interactions of a network [31].

The main differences of the used IGA from the conventional GAs are chromosome encoding and crossover operation mentioned above. Besides, the used mutation is also different from the conventional one, described as follows. Assume a real-value parameter x is to be mutated. A perturbation ¯x is generated by the Cauchy-Lorentz probability distribution [59]. The mutated value of x is x⁰ = x + ¯x or x − ¯x, determined randomly.

If x⁰ is out of the domain range of x, we randomly assign a feasible value to x⁰. The used simple IGA is described below.

Step 1: (Initiation) Randomly generate an initial population with N_pop feasible indi-viduals of 2(N + 1) real-value parameters.

Step 2: (Evaluation) Evaluate fitness values of all individuals.

Step 3: (Selection) Use the simple ranking selection that replaces the worst Ps× Npop

individuals with the best Ps×N_popindividuals to form a new population, where P_s is a selection probability. Let I_best be the best individual in the population.

Step 4: (Crossover) Randomly select P_c× N_pop individuals including I_best, where P_c is a crossover probability. Perform intelligent crossover operations for all selected pairs of parents.

Step 5: (Mutation) Apply the above-mentioned mutation operator to the population using a mutation probability P_m. To prevent the best fitness value from dete-riorating, mutation is not applied to the best individual.

Step 6: (Termination test) If a prespecified number N_eval of fitness evaluations is achieved or some stopping condition is met, then stop the algorithm. Oth-erwise, go to Step 2.