This paper deals with global optimization and focuses on the Self-Organizing Migrating Algorithm (SOMA). Three new versions of SOMA are proposed, where the first two introduce diferent mechanisms to maintain population diversity and the third combines both approaches. The algorithms were tested on the CEC2014 benchmark set at two levels of dimension, = 10
and = 30 . The results of the experiments suggest that two of the proposed modifications can improve the performance of the original SOMA-T3 variant in many cases.
CEUR Workshop ISSN1613-0073
SOMA (Self-organizing Migrating Algorithm) was introduced in 2000 by Zelinka and Lampinen [3]. The algorithm can be viewed as a model inspired by the cooperative hunting behavior of animal packs.
It works with a population of points from the search space. At the beginning of the run, the algorithm generates a random initial population and evaluates the objective (optimized) function in each of them. Then it works in cycles, so-called migrations. After each migration, the algorithm checks whether the stopping condition is met. When done, it returns the best solution found.
In each migration, each point travels towards the best point of the population which is named the leader. The point can also skip the leader. The length of the path depends on the parameters of the algorithm. Each point visits several places (positions) on its way to the leader. We can say that the point jumps on the line (its path). The best position of such visited places is the new possible position of this jumping point. The member of the population is moved to this new possible position if it is better than the original position of the member (in the sense of optimized function value). A migration includes such jumping and eventual moving for each member of population except the best point of population – the leader.
The movement of a member to the leader need not be straight. It can move only in some dimensions. There is a parameter which calls , this is a number between 0 and 1, more precisely the number of the interval [0, 1]. determines the probability that the movement of a point to the leader is done in a dimension. Before computing new possible positions for a population member, a is computed. It consists of elements, each element can be from the set {0, 1}. The relevant member is moved only in dimension(s) where has the value 1 in the corresponding position(s). Firstly, a vector of random numbers (from a uniform distribution) is generated. Then, where the vector has lower value than , the has number 1 at the corresponding place, at other places there are zeros.
The population member jumps onto the line determined by its position, the leader position, and
from its position to the leader position in steps. The is the algorithm parameter, a
number in the interval (
The next parameter of the algorithm is called ℎℎ . It determines the maximal length of the entire path from a population member to the leader. The maximum length is ℎℎ × ( − ) . The recommended value of ℎℎ ranges from 1.1 to 3.
Let us now look at how the new position is calculated in more detail. First, new positions are calculated for all members of the population during a migration. Then each point is moved in a single step to a better position (only in the dimensions selected by ), so that all points move together.
The last parameter is the size of the population , which is recommended to be set to 10, when the dimension of the solved problem is less than 100, otherwise it is recommended to have a population of 20 − 50 members.
The duration of the algorithm run is determined by the number of migrations or by the number of evaluations of the optimized function. [3] 2.1. Migration Strategies The authors of the algorithm proposed several diferent migration strategies. The one described in the previous paragraphs is called AllToOne. The other specification of the migration strategies is as follows: [3] • AllToOneRand: The leader is selected at random rather than based on the optimized function value here, introducing an element of stochasticity. • AllToOneAdaptive: In this migration strategy, the point is moved immediately after finding its better position (not at the end of the whole migration as in AllToOne migration). So for other members, the new position of the point can already be the leader if it is better than the previous leader. • AllToAll: Each point migrates to all other members of the population. The point is then moved to the best position of all its migration paths if the position is better than its original position. This strategy is computationally demanding. • AllToAllAdaptive: Similar strategy to AllToAll, but each individual immediately moves to a better position found during its migration path and then migrates to the next point of the population from its new position, instead of waiting (with moving) to find the best position of all its migration paths.
The SOMA-T3 algorithm is a very eficient modification of Self-organizing Migrating Algorithm. The
algorithm is described in detail in Algorithm (
It uses dynamic setting of parameters and . Both are set separately for each individual of
the population, the according to equation (
= 0.05 + 0.90 × ( / = 0.15 − 0.08 × ( / ), ).
Such a dynamic setting of the parameters results in the modification of only a few dimensions at the beginning of the run, and the jumps along the aforementioned line are relatively large due to a high , which supports a broad exploration. Later in the run, increases and decreases, so more dimensions are updated, but jumps are smaller. This helps the algorithm smoothly shift from exploring the search space to fine-tuning the best found solutions.
The algorithm does not use the parameter ℎℎ . It uses the parameter with the original meaning together with a parameter . The number indicates how many positions on the path from the original position of point towards the leader position (in some coordinates; with step × ( − ) , where is the position of the currently migrated point of population) the algorithm explores.
The SOMA-T3 algorithm also does not use only one population of points to solve an optimization problem. It uses a relatively large population, but it migrates only a part of this population, in each cycle the part is diferent. It has a population of points. For each migration, it randomly chooses the population of members from . And finally, only the best members of make the final population and migrate. Also, leader is not one of them. In this algorithm, members are randomly selected from to form the population of possible leaders. It is done separately for each migrant. The best point of is the .
We studied the SOMA-T3 algorithm and the goal was to improve its eficiency. Our first modification
lies in the following. We wanted to empower the exploration of the algorithm, but also to have a very
similar algorithm to the SOMA-T3 at the end of the search. We decided to add a tool that allows us to
extend the length of the whole path the point surmounts from its original position when it migrates at
the beginning of the algorithm and to cut this path at the end of the run. Originally we proposed the
number equal to × (1 − / ), it decreased from the beginning to the end of the algorithm run.
But such a number without any randomness may, of course, make a behavior which is not suitable for
all solved problems. So, we multiply this by the random number () between 0 and 1 and get the
ifnal from the equality (
g = × rand() × (1 −
) ,
(
which is the number of positions of a migrant on its path towards leader, here 2: Initialize
= 45. 3: Evaluate optimized function in all members of
solution of the problem 4: while <= do 5: = ∅ 6: ←
random selection (with uniform distribution) of migrant candidates from ← best migrants from
according to optimized function value for each from do ← random selection (with uniform
distribution) of leader candidates from ← best from
according to optimized function value
if == then
← second best from
according to optimized function value
according to (
according to (
Then, when the new positions of a migrant are computed, the vector, which was originally
added to the migrant in SOMA-T3 to get the new positions, see equality (
Mig = Mig + (Lea − Mig) ×j × STEP × PRTVec,
Mig = Mig + (Lea − Mig) ×j × STEP × PRTVec × g.
(
The algorithm SOMA-T3 with just described tool implemented in, we call SOMA-T3-RL (RL means something like random length).
Our second modification of the SOMA-T3 algorithm deals with another possibility of maintaining population diversity. In the original algorithm, the authors guarantee a suficient degree of diversity of the population by working with only a part of the population in each migration. Our approach lies in a smaller population and refreshment of some points when the search process stagnates.
We proposed changing randomly selected points of points (but not several best) of the population to a random position in the search space when the solution of the problem does not evolve in several last consecutive migrations. In this case, when the diversity of population is provided in this way, it is possible to work with a smaller population. And it is also suitable because of saving optimized function evaluations with regard to their depletion for members which position is changed, when it is needed.
So, we reduced the population size to less than about two-thirds and introduced three new parameters, ℎ , ℎ , . We are storing the last two solutions (of solved problem; the solution - the best point of population in a time) in each time and detecting if the last solution is or is not equal to the penultimate one. If they are equal for more than ℎ times, we change the position of ℎ randomly selected points of the population, but not the best points. This modification of the SOMA-T3 algorithm we call SOMA-T3-DIV in the following.
The third modification, called SOMA-T3-RL-DIV, merges the approaches from the first two variants into a single algorithm. It incorporates the random path length mechanism of SOMA-T3-RL, which adjusts the migration distance during the run, and the population refresh strategy of SOMA-T3-DIV, which replaces selected individuals when stagnation is detected. This combination allows the algorithm to use both strategies at the same time and potentially benefit from their joint efect.
We tested four algorithms: the original SOMA-T3 algorithm and three of its modified versions described above, SOMA-T3-RL and SOMA-T3-DIV, and also the combination of both modified variants SOMA-T3RL-DIV. In this initial stage of our experiments, we adopted the same parameter settings as used in SOMA-T3 to ensure a consistent baseline for comparison. For SOMA-T3-DIV and SOMA-T3-RL-DIV, we reduced the population size to less than about two thirds. This reduction is feasible because population diversity is maintained by randomly changing the positions of selected points while keeping the best points unchanged. As a result, with the same number of function evaluations, the search can be in some sense longer and can obtain better result.
The evaluation is carried out on the CEC2014 benchmark set [9], which consists of 30 functions designed for competition for single-objective optimization algorithms with real parameters. These functions include a mix of unimodal ( 1 − 3 ), simple multimodal ( 4 − 16 ), hybrid ( 17 − 22 ), and composition ( 23 − 30 ) types and ofer varying levels of dificulty and complexity.
Each algorithm was run 51 times for each configuration, with = × 10 4. In total, this resulted in 4 × 2 × 30 × 51 runs (count of algorithms × count of dimensions × count of functions × count of runs), that is, 12, 240 runs overall. The count of runs for a combination (algorithm, dimension, function) was chosen according to [9], and the dimensions 10 and 30 are also prescribed in it – in the future, we plan to test dimensions 50 and 100 as well.
All functions share the same search space, [−100, 100], for every dimension. The global minimum values range from 100 to 3000: 1 has a minimum of 100, 2 of 200, 3 of 300, and so on, up to 30 with a minimum of 3000. This makes to compute the diference between an algorithm’s solution and the true minimum straightforward. Therefore, in the results below, we report only this diference.
All four algorithms tested have several parameters. Some of these parameters are set the same across all algorithms: , , , and , with values = 10 , = 5 , = 15 , and = 45, following the original SOMA-T3 settings. Other parameters, such as ℎ , ℎ , and , are specific only to SOMA-T3-DIV and SOMA-T3-RL-DIV. The parameter difers for some of the algorithms tested. The specific settings of the last mentioned parameters are listed in Table 1. In this table, SOMA-T3-RL, SOMA-T3-DIV, and SOMA-T3-RL-DIV are marked as RL, DIV, and RL-DIV, respectively.
The testing was done on a MacBook Pro 14” with an Apple M3 Pro chip (12-core CPU, 18-core GPU) and 36GB of RAM. As a basis, we used the SOMA_T3A-python implementation (version 2), available on GitHub [10], modified to support the CEC2014 benchmark functions – together with Python version 3.13.2, NumPy version 2.3.1, and the Opfunu [11] library (with CEC2014 functions) version 1.0.4.
As a basis, we used a Python implementation of the algorithm sourced from GitHub [10], which we extended with the modifications described above.
In summary, we tested the three modified versions of the algorithm – by running each of them 51 times for the 30 CEC2014 benchmark functions, using the implementation provided by the opfunu library (version=”1.0.4”). [11] The same procedure was applied to the original unmodified algorithm, SOMA-T3. Moreover, all of these experiments were performed for both dimensions = 10 and = 30 , always using the parameters described above.
This section provides a detailed presentation and interpretation of the results obtained in our experiments performed on the 30 CEC2014 benchmark functions in dimensions 10 and 30. All results obtained were corrected as prescribed in [9]. If the result was less than 10−8, it was substituted by 0.
Some statistics calculated from the results of our experiments are printed in Tables 2, 3, 4, 5. In these tables, SOMA-T3-RL, SOMA-T3-DIV, and SOMA-T3-RL-DIV are also marked as RL, DIV, and RL-DIV, respectively. Tables 2, 3 summarize the comparison of each proposed algorithm (SOMA-T3-RL, SOMA-T3-DIV, SOMA-T3-RL-DIV) with the original algorithm SOMA-T3 in dimension = 10 . Tables 4, 5 show the summary of the same comparison in dimension = 30 . For each combination of dimension, function, and algorithm, we computed the minimum and median of the 51 results obtained. All minimums are shown in Tables 2 and 4. All medians are shown in Tables 3 and 5.
In these four tables, better statistics are underlined. We always compare a statistic (minimum, median) of a SOMA-T3 modification (SOMA-T3-RL, SOMA-T3-DIV, SOMA-T3-RL-DIV) with the relevant statistic of the original SOMA-T3 algorithm. That is the reason why the underlined numbers appear only in the last three columns. The last line in each of these four tables summarizes the count of wins and count of losses for each of the algorithms (SOMA-T3-RL, SOMA-T3-DIV, SOMA-T3-RL-DIV) when comparing the statistic in the dimension.
When we look at Table 2 (the numbers in the table are minimums in dimension = 10 ), we can
see that the SOMA-T3-DIV and SOMA-T3-RL-DIV algorithms have more wins (20, 21) than losses (
Without additional statistics, we can conclude that the proposed modifications SOMA-T3-DIV (which use a tool for preserving population diversity that difers from the one used by the original algorithm) and SOMA-T3-RL-DIV (an algorithm in which we implemented both proposed tools) of the SOMA-T3 algorithm are suficiently efective. SOMA-T3-RL does not appear to be as efective as expected. Here we can observe the diference in the improvement of efectiveness when randomness is restricted to a single direction, and when it is not restricted and diversity is supported by points across the whole search space.
In order to know the true diference in the efectivity of the original algorithm and the proposed algorithms, we computed 180 Wilcoxon rank-sum statistical tests. All statistical tests were performed at the significance level set at 0.05. The results of the statistical tests performed are shown in Table 6. In this table, RL indicates SOMA-T3-RL, DIV is SOMA-T3-DIV, and the label RL-DIV means SOMA-T3-RL-DIV. F When + appears in the table, it means that the respective modified algorithm is statistically better than the original, − means that the original algorithm works statistically better than the modified one, and ≈ means that both algorithms compared work with the statistically same efectivity.
When we compare the original algorithm SOMA-T3 and SOMA-T3-RL in dimension = 10 , we found that only for the 2 functions tested the results of SOMA-T3-RL are statistically better than the results of SOMA-T3, and simultaneously the results of SOMA-T3-RL are statistically worse than the results of SOMA-T3 for 10 functions of the benchmark set. So, the conclusion should be that SOMA-T3 is better than SOMA-T3-RL in = 10 . In dimension = 30 , the situation is very similar. SOMA-T3 is statistically better than SOMA-T3-RL 13 times and SOMA-T3-RL is statistically better than SOMA-T3 6 times. The following conclusion should be drawn. The SOMA-T3 algorithm is more efective than the SOMA-T3-RL algorithm also in dimension = 30 .
When we look at the columns for the comparison of SOMA-T3 and SOMA-T3-DIV, we can see that the numbers at the bottom of the columns are diferent. In dimension = 10 , the SOMA-T3-DIV algorithm is better than the SOMA-T3 algorithm for 12 functions. On the other hand, the SOMA-T3 algorithm is more efective than the SOMA-T3-DIV algorithm for only 5 functions from the benchmark set. A similar situation appears for dimension = 30 . SOMA-T3-DIV is better than SOMA-T3 for 16 functions and SOMA-T3 wins only on the 6 functions tested. Looking at the last two columns of the table, we can conclude that the combined use of both proposed tools does not yield significantly F diferent results compared to the implementation of only the DIV mechanism. The SOMA-T3-RL-DIV wins 17 times in dimension = 10 and 12 times in dimension = 30 . On the other hand, the last comparison looks as follows. SOMA-T3 wins 4 times and 6 times in dimension = 10 and = 30 , respectively.
When we summarize the comparison of SOMA-T3 and SOMA-T3-RL, the part of functions where SOMA-T3-RL is statistically better than the original algorithm is equal to or less than 20 % in each tested dimension. However, when we think about the two other comparisons – SOMA-T3-DIV and also SOMA-T3-RL-DIV, we find that SOMA-T3-DIV and SOMA-T3-RL-DIV are more efective than the original SOMA-T3 algorithm for more than half of the functions in a given tested dimension: SOMA-T3-DIV in = 30 and SOMA-T3-RL-DIV in = 10 . In the other dimension, they are both better than SOMA-T3 in more than 33 % of the tested functions. Overall in both dimensions together, SOMA-T3-DIV and also SOMA-T3-RL-DIV are worse than the original SOMA-T3 algorithm only in about 16 % of the benchmark optimization problems. The results of these two algorithms are of similar or better efectivity than the original SOMA-T3 in the rest of the benchmark problems, that is, in more than 80 % of the tested problems. F
In this article, we focus on the SOMA optimization algorithm, in particular the SOMA-T3 variant, which we modified to improve optimization performance. Three algorithms based on SOMA-T3 were proposed: SOMA-T3-RL, SOMA-T3-DIV, and a combination of both, SOMA-T3-RL-DIV.
The results show that SOMA-T3-RL (with random length of the migration vector) does not bring significant improvements and in some cases leads to a slight decrease in performance compared to the original algorithm. However, its performance remains largely comparable to that of the original SOMA-T3, suggesting that this modification does not introduce instability or consistent deterioration.
In contrast, SOMA-T3-DIV (with a mechanism for maintaining population diversity) demonstrates stronger and more robust improvements in both dimensions tested. The combined variant, SOMA-T3RL-DIV, also achieves competitive results, reaching a similar or even better balance of improvements compared to SOMA-T3-DIV.
In general, the proposed modifications confirm that the most efective strategy is to enhance population diversity (DIV). The combined variant SOMA-T3-RL-DIV provides results similar to SOMA-T3-DIV and in some cases is even slightly better, showing that the random length mechanism can bring a small additional benefit when used in combination with population diversity maintenance.
However, a space for further improvement remains there, suggesting that additional modifications or
This work was supported by the Silesian University in Opava under the Student Funding Plan, project SGS/9/2024.
Either: The authors have not employed any Generative AI tools.