Population Migration Algorithm

Consider the following unconstrained single-objective optimization problem:(1)

In which is a real valued mapping, is a search space, . We assume that problem (1) has a constant solution, namely, global optimal solution exists and the set H of the global optimal solution is not empty. Seeking the most value problem can be transformed into the form of problem (1). In the algorithm, is an individual and stands for the population and various kinds of information. is the ith individual in the solution space, ; is the ith individual of the jth weight; , is the jth weight of the ith region’s radius , in which, , , ; and N stands for population size. Target function f(x) corresponds to the population attraction. The optimal solution (a local optimal solution) of problem (1) corresponds to the most attractive region (the preferential). The ability of the PMA to escape from the local optimal solution corresponds to the ability. Due to population pressure increase, the population moves out of the preferential area. The rise of algorithm or mountain climbing made the populations move into the preferential area. Local random search of algorithm corresponds to the population flow. The method by which the algorithm selects the approximate solution corresponds to the population migration. Escaping from the local optimal strategies corresponds to the population proliferation.

Frog Leaping Algorithm of Local Search

SFLA is proposed, as proved by Eusuff in 2003 [9]. SFLA is based on the heuristic and collaborative search of species. The implementation of the algorithm simulates the element evolution behavior of nature.

The implementation process of SFLA simulates the feeding behavior of a group of frogs in a wetland [10]. In the feeding process, every frog can be seen as the carrier of ideas and information. Frogs can communicate through the exchange of information, which can improve the knowledge of others. Every frog represents a solution to the optimization problem (1). Wetland represents the solution space. In the algorithm implemental stage, there are F frogs divided into N groups. Each group has l frogs. Different groups have different ideas and information. In a group, frogs follow the element evolution strategy in the solution space for local depth search and internal communication.(2)(3)

In the above formulas, rand() is a random function that randomly generates the value in the range [0,1], S is a step of the frog leaping, P_wi stands for the worst frog of the ith group, P_bi stands for the best frog of the ith group, and P1_wi stands for an update of the worst frog of the ith group. If the adaptive value of the updated worst frog is more than that of the original worst frog in the same group, the updated worst frog will replace the original worst frog. Otherwise, a frog will be randomly generated in the group instead of the original worst frog. The updated process is repeated until the predetermined number of local search L_S. When all groups have completed the depth of local search, all the frogs of the groups are again mixed and divided into N groups. We continue the local search in the groups until the termination rule holds true, and then we stop.

Crossover Operators

By two leaner combinations, we can obtain two new individuals. Arithmetic crossover operation is often used to express individuals by floating individual coding. Hypothesis: The two parents are and .

Then, n random numbers are randomly generated, in which , . After hybrid implementation, two new progenies are generated:

Initialization

Input the population size N, the initial radius δ(0), the vigilance parameter of population pressure α(0), the number of the population flow l, the number of local search L_S, and the maximum number of iterations T. We randomly generate N individuals in the search space S, X¹, X², …, X^N. The ith regional center determines the upper and lower bounds of the ith region, in which , , .The extraction method makes equal, so we cancel the superscript of in the following steps. We obtain an initial search space , which stands for the initial residential area. The set is taken as the sphere, in which X is the center and r is the radium. The following variables appearing that way are the same. and . We set iteration counter t = 0.

Evolutionary

1. Preparation:
2. Population flow: In every , we randomly generate l_i individuals, so we can obtain the group in , which has Nl individuals.
3. Population migration
   We next introduce the local search mechanism of frog-leaping algorithm and crossover operator. At this time, the role of population has been transformed to the frog. The search area is seen as the frog’s wetland. The formed N local areas are seen as the N groups. The individual of every local area is seen as the frog.
   1. (a) We choose N best frogs in Y(t), which form the intermediate frog group .
   2. (b) Forming N regions: .
   3. (c) In every group, we produce a frog body flow. In other words, l_i individuals are randomly generated in . l_i and the adaptive value of are proportional.
   4. (d) We set the group counter iN = 1, which is compared with the total of groups N; the local evolution counter im = 1, which is compared with the number of local search L_S.
   5. (e) In the iNth group, we calculate the adaptive value of every frog in the group and arrange them in descending order. We can obtain the best solution P_bi and the worst solution P_wi.
   6. (f) Ph_wi = P_wi, according to the following four formulas, we can update and improve the position of the worst frog in the group.
      (4) (5) (6) (7)
   7. (g) If step (f) improved the position of the worst frog, the adaptive value of is more than the P_wi’s, the position of P_wi will be replaced by the position of .That is, . Otherwise, we randomly generate a position of a frog instead of the worst frog in the group.
   8. (h) If im<L_S, then im = im+1and turn to step (f). Otherwise, we report the best frog and turn to step (i).
   9. (i) If iN<N, then iN = iN+1 and we turn to step (e). Otherwise, we turn to step (j).
   10. (j) Contraction of preferential region:.
   11. (k) If (not more than the population pressure alert), we continue the frog flow and turn to step (c) Otherwise, we turn to step (l).
   12. (l) We report N the best frogs:.
4. Population proliferation:
   1. (a) We keep the best individual in the , which is credited as .
   2. (b) In the , we randomly sample N-1 new individuals instead of the N-1 individuals of .
   3. (c) We define a new generation of population

Termination of Inspection

If the new generation of population X(t+1) contains the eligible solution, we stop and output the best solution in X(t+1). Otherwise, we narrow and properly and set t:t+1. If the maximum numbers of iterations for the algorithm are not met, we turn to the second step.

Convergence Analysis

This paper uses the axiom model to analyze and prove the convergence of IMPA.

Definition 1 Optimization variables X have limited-length characters and have the form . n is called the code string length of X, A is called the code of X, and X is called the decode of the character A. a_i is considered a genetic gene. All possible values of a_i are called allelic gene. A is a chromosome made up of n genes [11].

Definition 2 (Individual Space) is an allelic gene. n is given the code string length. The set is called individual, in which is a natural number and is the order of population space [11].

Definition 3 (Satisfaction Set) If , then . B(X) is called the satisfaction set. The number of individuals in the set B(X) is used for . is any of the population and is the best individual in the set . The satisfaction set induced by is called satisfaction set of the population, which is marked . All of the satisfaction sets’ intersections are formed with the global optimal solution [11].

Definition 4 (Selection Operator) The selection operator is a random mapping, . It meets the following two conditions. (1) For , we can obtain , in which stands for the individual in the ; (2) Any population that meets has . The first condition states that selected individuals should be in the selected population. The second condition states that the selection operator may increase the number of optimal individuals in the population, in addition to variations of the individuals’ numbers between selected population and original population [11].

Definition 5 (Reproduction Operator) Reproduction operator is a random mapping.. If any population , which meets , the new population which becomes under the action of the operator, may contain more satisfied individuals than the original population [11].

Definition 6 (Mutation Operator) Mutation operator is a random mapping: .meets. Mutation operator is the reproduction operator that has a much better nature. If the population contains satisfied individuals, then after mutation, the population should contain satisfied individuals. If the population does not have satisfied individuals, then after mutation, the population may contain satisfied individuals [11].

In improved population migration, extended operator meets the following expressions:and

Therefore, we can use the operator to stand for the process of population flow. The selection operator S₁ stands for the process of step (a) in population migration, where we chose N, the best individuals from population Y(t). The reproduction operator R₁ stands for the progress from step (b) to the process of final output Z_best(t) in step (l). The reproduction operator R₂stands for the process from population proliferation to finally generate X(t+1). Therefore, the IPMA can be expressed as:(8)(9)

After both sides have the same function of operator for (8), we can find that .The general simulated evolution algorithm can be composed of a series of selection operators and reproduction operators to be represented abstractly as follows: .

Lemma 1 In the IPMA,S is a selection operator and R is a reproduction operator. The proof can be obtained from reference [11].

Theorem 1 The IPMA is a simulated evolutionary algorithm.

Lemma 2 In the IPMA, the characteristic number of the preferred selection operator, the selection pressure of the operator S:P_S = 1, the selection intensity α_S = 1. The IPMA just takes the preferred selection operator. According to reference [3], the selection pressure and the selection intensity are defined to calculate the characteristic number of the operator S. The proof can be obtained from reference [11].

Lemma 3 In the IPMA, the characteristic number of the reproduction operator, the operator , then the absorption and scattering rate of the reproduction meet the following estimates: . Where, (is the radius of the search space) is the radius of the set of optimal solution and is a conversion coefficient. If the set includes a division region in the , then; otherwise, is the rate of volume of the intersection of the set and a division region. Thus, we can obtain . The proof can be obtained from reference [11].

Lemma 4 If the abstract simulated evolutionary algorithm (SEA) satisfies the following conditions: (1) the pressure of selection operator is uniformly bounded; (2) the absorption of reproduction operator: meets ; (3) the strength of selection , absorption , and scattering content are as follows: . Then, an abstract and a general SEA will probably weakly approach the optimal solution set. The proof can be obtained from reference [11].

Theorem 2 (Convergence of IPMA) If the IPMA adopts the selection operator family , which comprise all the preferred selection operators, reproduction operators family, in which . The following conditions are thus met: (1); (2) ().

Then, to prove that the IPMA will probably weakly approach the solution set, the following steps are performed.

Simulation Examples and Experimental Parameter Setting

To verify the performance of the IPMA, this paper performs an experiment. The IPMA is compared with the basic population algorithm and other improved algorithms. In the simulation experiment, we choose 10 classic test functions to test the algorithm. According to the performance of the functions, they are divided into a single minimum (single) and a number of local minima (multimodal) into two categories. The following lists the 10 functions’ definition, variable range, and global optimal values of the theory and optimal solution of the theory. The function f₁ is a unimodal function used to examine the accuracy of the algorithm. The function f₂ which contains infinite local optimal solution and has infinite local maximum near suboptimal solution, is a strong shock multimodal function. General optimization algorithms easily fall into local optimal solution. Thus, the ability of the algorithm is used to avoid falling into a local optimization algorithm. The function f₅ is a multimodal function. It has about 10*D (D is dimension) local optimal solution. The function f₆ is a deep and local minimum multimodal function that has many local optimal solutions. The functions f₅ and f₇ are complex nonlinears used to test global search performance of algorithms. The function f₈ is a complex and classical function used to test the optimization efficiency of algorithms. In a certain number of iterations, this paper estimates the performance of IMPM by the test function of the best value, the worst value, average value, variance, average running time, and convergence rate.

1. Sphere Model function: . The optimal solution is and the optimal minimum value is 0, when n = 30.

2. Schaffer function: , , i = 1,2. The optimal solution is (X₁, X₂) = (0, 0) and the optimal maximum value is 1.

3. , , . The function in the section (−5.12, −5) has a global minimum value −30.

4. , , . The optimal solution is and the optimal minimum value is −6.

5. Generalized Rastrigin function: . The function has many local minimum solutions, but it has only one global minimum solution (X₁, X₂) = (0, 0) and the global minimum value is 0, when n = 2, n = 5 and n = 10.

6. Quartic function: ,, . The optimal solution is and the optimal minimum value is 0, when n = 20.

7. Ackley’s function: , . The optimal solution is and the optimal minimum value is 0, when n = 20.

8. Generalized Rosen Brock’s function:

. The optimal solution is and the optimal minimum value is 0, when n = 3.

9. Griewank function: . The optimal minimum value is 0, when n = 30.

10. Schwefel function: . The optimal minimum value is −2094.9, when n = 5. The optimal minimum value is −20949, when n = 50.

To verify the performance of the IPMA, we choose a computer whose CPU is made by AMD and has an operating system of Windows 7. The mathematical software Matlab7.0 was used for the testing. Each function is applied 50 times. We present the parameters in the process of solving the functions for this paper in the table 1. To show the better performance of the IPMA, we compare it with other algorithms through the following indicators: best and worst solutions, average solution, variance, average running time, and average convergence rate in 50 cycles.

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Table 1. Parameter setting of the functions for the IPMA.

https://doi.org/10.1371/journal.pone.0056652.t001

This table shows parameter setting of the ten functions. Population size N, number of population flow l, radius δ, coefficient of contraction Δ, pressure of population parameters α, number of local search L_S, and largest number of iterations K. .

Experimental Result and Analysis

In Table 2, the data of function f₁ show that the improved population algorithm is better than frog-leaping algorithm in terms of precision. Other algorithms find it very difficult to achieve the optimal solution of function f₂, but the IPMA can easily do so, which indicates that the algorithm’s ability of escaping from local optimal solution is very ideal. Function f₈ can test whether the algorithm’s ability of global optimization is good. It is used to measure the efficiency of the algorithm, so we know the efficiency of the IPMA is very high compared with other algorithms from the data of function f₈.The data functions f₃, f₄, f₅, f₇, f₉ and f₁₀ reflect that the IPMA is a better algorithm from a certain angle. Through cooperation of the best value, the worst value, the average value, the variance, and the convergence rate, the IPMA is found to be better than the other algorithms in the calculation of stability and precision.

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Table 2. Comparison performance of the IPMA in the ten functions.

https://doi.org/10.1371/journal.pone.0056652.t002

Conclusions

In this paper, based on the local search mechanism and crossover operator, we put forward an IPMA. The parameters of the IPMA are simple and easy to realize. By testing the 10 functions, we find that the IPMA is better than the basic PMA and is superior to many intelligent algorithms. The PMA is good at global searches, but it does not perform well in local search and its precision is low. We introduce local search in population and crossover operator to improve these deficits. Therefore, the IPMA is increased to avoid falling into local optimum capacity; its calculation speed and precision are also improved.