In this paper, we focus on the ongoing research of the Cooperation algorithm, which combines features of the established and successful GWO and jSO optimization algorithms. We introduce a new enhancement using the stagnation parameter, which ensures switching between the two algorithms when no further improvements are achieved during the optimization process. We test diferent settings of the stagnation parameter and find its efect on the performance of the Cooperation algorithm.
1. Introduction
vironment. The environment is represented by a real In this paper, we deal with two well-established and function instead of our planet in GWO. The movement of successful optimization algorithms, namely jSO [2] and the agents simulates the movement of real grey wolves GWO [3]. We have introduced the earliest experiment on in their search and hunt for prey and their cooperation the cooperation of these two methods in [1]. The results with each other. of that experiment were promising, which motivated us Let us focus on the hierarchy in the wolf pack. A pair of to continue our research by modifying the proposed al- Alpha wolves are the highest in the hierarchy, are pack gorithm and testing it on more functions. The progress leaders and have reproductive duties. The Betas support and results of the continuing research will be presented their decisions and give them feedback. Deltas take care in this paper. First, we briefly introduce the two main al- of routine tasks - protecting the pack in case of danger, gorithms. Then we will introduce a new method for their taking care of old and sick wolves, assisting with hunting, cooperation, test it on functions of CEC2014 benchmark etc. Omegas are the lowest in the hierarchy, for example set [7], and evaluate the efectiveness of the proposed they can eat last and other wolves can take their frustraapproach. tions out on them, which helps keep the pack stable. The hierarchy is also applied in a simplified form in the 2. GWO and jSO GWO algorithm. The point in the environment at which the wolf is currently located has a concrete value comGWO and jSO are well-known, successful, and well- putable by a fitness function. The fitness function comestablished optimization algorithms. Like many other putes the value of the function at the position of the agent optimization methods, they are stochastic, heuristically that corresponds to a possible solution to the optimizaderived, and inspired by nature. tion problem. The 3 wolves with the best fitness function values at the moment are labeled as Alpha, Beta, and Delta. From the point of view of GWO, they are equal. 2.1. GWO And all other wolves are marked as Omega. GWO or also called Grey Wolf Optimizer, was introduced Another important characteristic of the wolf pack that in 2014 by S. Mirjalili and his collaborators [3]. It found GWO is inspired by is the method of hunting. In the real inspiration in the hunting and social hierarchy of the world, wolves hunt in the following phases: searching wolf pack. The GWO algorithm is multi-agent system. for prey, stalking and closing in on it, encircling it, and once the prey is encircled, coming into attack to the weak area. GWO distinguishes between a Searching for prey 23rd Conference ITAT (Information technologies – Applications and phase, when wolves tend to move away from the best Theory) solution found so far to the optimization problem, and * Corresponding author. a Hunting phase, when they move closer to the "prey" † These authors contributed equally. (currently best position found so far, based on positions $ daniel.valenta@fpf.slu.cz (D. Valenta); of agents Alpha, Beta, and Delta). radka.polakova@fpf.slu.cz (R. Poláková) Agents, like wolves, move around the environment with (R. 0P0o0l9á-k0o0v0á5)-0781-7755 (D. Valenta); 0000-0002-0782-5668 the goal of finding the best possible food. Because it © 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License is an iterative algorithm, in each iteration each of the CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g ACttEribUutRion W4.0oInrtekrnsahtioonpal (PCCroBYce4.0e).dings (CEUR-WS.org) wolf-agents moves to a new position. Before each move, agents decide whether to hunt (Hunting phase) for the best prey found so far (it moves in direction to currently the best result found), or to further explore the environment (Searching for prey phase) where even more abundant prey may be found (the truly global optimum).
Which phase each agent chooses depends on two random vectors, ⃗ and ⃗. Each agent has both vectors assigned to it. The vector ⃗ has components (0, 2), where (0, 2) generates a random number between 0 and 2 with uniform distribution. The vector ⃗ has components (− 1, 1) × , where (− 1, 1) generates a random number between − 1 and 1 with uniform distribution and = 2 − (2/), where is the current iteration of the algorithm and is the maximum number of iterations of the algorithm, also specified as a termination criterion. The components of both vectors afect the movement of the agent in each dimension of the environment. The closer the value of the component is to 0, the more probably the wolf is to hunt in a given iteration (chooses the Hunting phase), and the closer it is to 2, the more likely the wolf is to explore the environment elsewhere (chooses the Searching for prey phase).
Vector ⃗, unlike vector ⃗, depends on the current iteration of the algorithm and ensures that wolves explore the environment more in the initial iterations (with higher probability) and hunt towards the end of the algorithm.
The vector ⃗ is purely random and simulates various obstacles in the environment, analogous to nature. As a result, agents tend to choose both phases independently of the iteration of the algorithm. This is needed to reduce the probability of converging to a local optimum instead of finding a global one in later iterations.
Now, we put both vectors into the context of calculating the new wolf-agent position. This is calculated as follows: ⃗ ( + 1) = ⃗1 + ⃗2 + ⃗3 3
, where is an agent with index , is the current iteration of the algorithm, and ⃗1, ⃗2, and ⃗3 represent potential new positions of the agent, which are calculated based on the positions of the three best Alpha, Beta, and Delta wolves as follows: 1 = ⃗ () − 1 × ⃗ , ⃗ ⃗ 2 = ⃗ () − 2 × ⃗ , ⃗ ⃗ and Delta, and ⃗ , ⃗ , and ⃗ represent the distance of the agent from the prey, but due to the efect of randomness of instances of vector ⃗, they are an approximate estimates, and are calculated as follows: = |1 × ⃗ () − ⃗ ()|, ⃗ ⃗ = |2 × ⃗ () − ⃗ ()|, ⃗ ⃗ = |3 × ⃗ () − ⃗ ()|, ⃗ ⃗ where ⃗1, ⃗2, and ⃗3 are three diferent instances of the vector ⃗, for each of the agents Alpha, Beta, and Delta, and ⃗ () is the position of the wolf in the current iteration of the algorithm.
The symbol × in the calculations of ⃗1, ⃗2, and ⃗3 and ⃗ , ⃗ , and ⃗ represents multidimensional vector multiplication by components. For example, for = 2, the calculation is performed as follows: (1, 1) * (2, 2) = (1 * 2, 1 * 2).
Having described all the principles, we now describe the sequence of steps of the algorithm. The input of GWO is the environment definition, in our case we use the CEC 2014 functions. The next input is the number of agents (pack size). The next and last input is the termination criterion, which can be defined by the maximum number of iterations, or alternatively by the maximum running time of the algorithm in the number of computations of the objective function value.
The pseudocode of the algorithm is as follows:
agent is located, the fitness function is calculated, 2. based on fitness values, agents are ranked in a hierarchy: the agent with the best value becomes Alpha, the second best Beta, the third best Delta, and all others Omega, 3. vectors ⃗ and ⃗ with random components are
generated for each agent , 4. it calculates new position of each wolf ⃗ in
search space, 5. finally, the algorithm checks to verify that the termination criterion - usually the expiration of the maximum number of iterations - has already been met; if so, the algorithm ends; if not, the algorithm continues with the first step.
2.2. jSO ⃗3 = ⃗ () − ⃗3 × ⃗ , The jSO algorithm is a successful population-based optimization method that is an improved version of the iL-SHADE algorithm [5] derived from diferential evo⃗ (), ⃗ (), and ⃗ () are the positions of agents lution DE [4]. Its improved strategy uses the weighted Alpha, Beta, and Delta in the current iteration of the mutation current-to-pbest/1 /bin and other tools like an algorithm, ⃗1, ⃗2, and ⃗3 are three diferent instantia- archive to store solutions which were rewritten in poptions of the vector ⃗, for each of the agents Alpha, Beta, ulation by better points and circle memory for storing parameters of distributions for generating new values of 0 = (⃗1, ⃗2, ..., ⃗NP ). At this point, the archive is DE parameters computed based on successful values of empty, so = ∅. Also, for each point in the initial poputhese parameters. lation 0, the value of the objective function is calculated. An equally interesting feature of jSO is that its population Finally, the default values of the parameter settings are size is linearly reduced like in its predecessors L-SHADE set during initialization. [6] and iL-SHADE [5]. Behind the success of jSO is also As another tool, the jSO algorithm uses two circle memthe appropriate setting of its parameters. We describe ories: for storing the parameters of Cauchy distributhese principles in detail in the following text. tion for generating values of mutation parameter and The jSO is an evolutionary algorithm that works with a CR for storing the parameters of normal distribution population that is evolving. It is an adaptive variant of for generating values of crossover parameter . The the Diferential Evolution (DE) algorithm. The algorithm size of these circle memories is = 5 and they have all works with a population of points that evolve (move to values set to 0.5 when the algorithm is initialized. diferent positions in the search space) by several oper- The jSO algorithm works as described in the following ations, mutation, crossover, and selection. It uses the pseudocode. After initializing the population points and already mentioned current-to-pbest-w/1 strategy as a mu- the parameters listed above, the while loop follows these tation strategy for selecting a so-called trial point, which steps: is a potential new point for the next generation. The 1. For each point generate a random number in the advantage of this strategy is that it adaptively controls range 1, 2..., , where is the size of the circle its parameters and that it uses the archive. Thanks to memory. If = , set both CR and to this mutation strategy, the algorithm selects one point 0.9. If CR is not greater than or equal to 0, set from the top few in the population to work with it and to 0. And finally, if CR > 0, generate also can use a point from the archive. The formula is as using normal distribution ( , 0.1). follows: 2. For each point, generate parameter using ⃗, = ⃗, +(b⃗est, − ⃗,)+(⃗1, − ⃗2,), 3. ICfatuhcehcyudrriestnrtibnuutmiobnero(feva l,u0a.t1i)o.ns is less than quarter of the maximum number of evaluations, the probability is limited to (, 0.7) for the point, and if the number of evaluations is less than half, is limited to (, 0.6) for the point. 4. If the current number of evaluations is less than six tenths of the maximum number of evaluations, the new value of parameter is limited to (, 0.7) for the point. 5. For each point ⃗ from population, a new trial point ⃗ is created using DE/current-to-pbestw/1/bin strategy, and then the value of the objective function in ⃗ is computed. 6. If (⃗) is better than (⃗), update the point ⃗ by ⃗ for further evolution. Otherwise, discard the trial point ⃗ and keep the original ⃗. 7. If point ⃗ is updated (by trial point ⃗ in the previous step), put it in the archive . If necessary, shrink archive . Also insert value of parameter into and into . 8. After doing steps 1-7 for all points in population calculate the new value of the first parameter for both distributions used for generating and from and . Then, put them into and where ⃗, is the − ℎ point of generation . , are the parameters of the point ⃗ from interval [0, 1] that determines the weights of the diferences in the mutation, ⃗1 , ⃗2 are two diferent randomly selected points, the first one is randomly selected from the population in generation , and the second one is randomly selected from the union of population and the archive, and b⃗est, is one of the × best points in generation (randomly selected), where is the size of population and ∈ [0, 1] is a dynamic parameter randomly afecting the number of points from which one is selected for this mutation (this helps to balance between exploitation and exploration phases).
The strategy current-to-pbest-w/1/bin additionally includes the binomial crossover operator. The parameter of this operator is adapted randomly. It is the probability that a component of the mutant ⃗, will be used in the trial point. is a random number with normal distribution whose first parameter (mean of the distribution) depends on successful values of used before.
Each component from ⃗, is written into the trial point with probability , when the component does not write, relevant component of original ⃗, is used. If the trial point is selected as a new point for the next generation, the original ⃗, is stored in archive .
More information about this mutation strategy can be found in [2].
The algorithm has a lot of input parameters that need to be set during the initialization phase. First, an initial generation 0 of size NP is randomly generated, i.e.
CR . 9. Increase the pointer in the circular memory by
1, and if it is greater than , set = 1 again. 10. Apply linear population size reduction mechanism and update parameter for mutation current-to-pbest-w/1. 11. If the termination criterion is not met, the com- dom points of the jSO algorithm except for the top three, putation continues by repeating the cycle for the which are labeled as non-rewritable and are kept as reprenew generation = + 1. Otherwise, the sentatives of the current best solution. Now, we describe computation terminates and the result is the best the switching of the jSO algorithm to GWO. The GWO point in population at current generation . algorithm reads the 6 best points (generally the number corresponding to the size of its population, ) from the jSO output and replaces all existing points with them. If 3. Cooperation algorithm the output of jSO (the population is continuously decreasing) contains fewer points than need to be replaced in As we have already mentioned, both algorithms GWO GWO when the algorithm is switched (i.e. the jSO popuand jSO are very useful for solving optimization problems lation is smaller than the GWO population), then only the and provide very good results in solving them. According maximum possible number of points from jSO is replaced to the No free Lunch theorem [8], it is not possible to say in the GWO population. Which points are replaced in which one is better, each is suitable for diferent types of the GWO population is chosen randomly, however the tasks, which is confirmed also by our testing results on best point in the GWO (Alpha agent position) is always CEC 2014 functions. For some functions, GWO provides kept. better results, while for others jSO provides better results. The next issue is how to properly determine after what The results of both algorithms are shown in Tables 1 and time the algorithms should switch to another one. In 2. We wondered how to take advantages of both of them. previous research, we changed algorithms always after a And so we introduced the first version of the Cooper- static number of evaluations. Each algorithm participated ation algorithm using the findings of GWO and jSO in equally in the solution. That is what we improved, and [1]. The principle of Cooperation is based on switching for this research, we decided to choose a new strategy both GWO and jSO algorithms. Already in this first early of switching between both algorithms after a constant version, which we tested only on a few functions (opti- number of generations/iterations when the algorithm mization problems), the results were promising. Based is already stagnant and no further improvements are on the results, we proposed further modifications to the achieved. Specifically, we test the algorithms to switch Cooperation and tested the improved algorithm on all after 30, 60, 90, and 120 iterations (for GWO) or generathirty CEC 2014 functions [7] at two levels of dimension. tions (for jSO) after it starts to stagnate. This allows us In this section, we describe the principles of its working to find the optimal running time of the algorithms and and present new adjustments. In the following sections, outline a way for further research, in which we try to we evaluate the performance of the new proposed Coop- change the running time dynamically after stagnation. eration in relation to the testing results. When the algorithm stagnates, before passing the populaLet us focus on the principles of the Cooperation algo- tion to the second algorithm, we bring the configuration rithm first. We run one of the original algorithms (GWO to a state just before stagnation. The positions of the or jSO) for a limited time, let it solve the problem for agents (for GWO) or points (for jSO) at this moment are some time, and pass its result as input to the second al- passed to the second algorithm as input. This improvegorithm (which has not yet run), which we also run for ment ensures that the second algorithm starts from a a limited time to solve the problem for another piece of better starting position, reducing the probability of rapid time. stagnation immediately after execution. This simple idea allows us to use all advantages of both Which algorithm starts the computation first is randomly algorithms, but in fact it is a bit more complicated. Both selected. Each of the algorithms has specific advantages, algorithms work with diferent population sizes (number and by randomly selecting the first one to run, we ensure of agents in the case of GWO and number of points in independence of their characteristics. population in the case of jSO). To simplify the following The flow of the Cooperation algorithm is shown in Figure text we will use a single term - a "point" for both algo- 1 and proceeds as follows: first, an initial population of rithms, and in GWO, we can imagine by the term "point" points (or agent positions) is randomly generated for the the position where the agent is located. ifrst algorithm, which is randomly selected from GWO In GWO, we use the usual number of points (agents) - and jSO. Then, the following steps are performed in a cywhich is 6. This number remains constant during the cle until the maximum number of evaluations is reached: whole calculation of the algorithm. While in the jSO the number of points is dynamically changed during the op- 1. If the GWO algorithm is selected, let it run until timization, the population size decreases linearly. there is no further improvement in the results First, we describe the switching of the GWO algorithm and thus the algorithm stagnates for more than to jSO. The output points of the GWO algorithm are iterations, where in our case is set to 30, 60, 90, read into the jSO algorithm and they replace the ran- or 120.
Once the GWO run is interrupted due to stagnathe used optimization algorithms (jSO, GWO and Cooperation) and are defined in multiple dimensions, where the number of dimensions is chosen by us. We tested a total of 6 algorithms: GWO, jSO, and Cooperation with the stagnation parameter set to 30, 60, 90, and 120. Testing was performed at two levels of dimension, = 10 and = 30, and each configuration was run 15 times for each of the 30 test functions.
In this section, we focus on the results of testing the Cooperation algorithm with diferent settings of the stagnation parameter ( = 30, = 60, = 90, and = 120), compare it with the original algorithms, and derive conclusions. 4.1. Test results in dimension = 10
The results of testing the algorithms in dimension = 10 on all CEC2014 problems (functions) are shown in
Table 1. In the columns GWO and jSO, we can see the tion, bring the configuration to a state just before median results of 15 runs of the original algorithms with stagnation. The random points in the jSO input these names. The best among these two medians is shown are overwritten by the points from the GWO, in bold. The fact that jSO gives better results in the mediwhere is the population size of the GWO. ans for all 30 functions is interesting. In the first iterations However, the best 3 points in jSO population can- of the computation, the GWO results seem promising, but not be overwritten. it converges too early and if we let both algorithms run 2. If the jSO algorithm is selected, let it run until long enough (in our case, the total amount of allowed there is no further improvement in the results function iteration (in GWO) or evaluations (in jSO) is and thus the algorithm stagnates for more than 10000 × for each dimension ), jSO finds a more acgenerations, where in our case is 30, 60, 90, or curate result (closer to the optimum) in the median of 15 120. runs.
Once the jSO run is interrupted due to stagnation, The following columns show the results of the Cooperabring the configuration to a state just before stag- tion algorithm with diferent settings of the stagnation nation. parameter , which is given in corresponding column. The best jSO points ( is the population size For example, Coop 30 means that a Cooperation algoof the GWO) replace each of the original GWO rithm with stagnation parameter = 30 was used, and points and are used as its input. similarly for the other columns of the table. Each value recorded in that table is the median of 15 results of 15 runs of respective algorithm (or setting of Cooperation 4. Test results algorithm) for specific problem (function). Value of the median which is better than values of the median of both The CEC2014 competition introduced benchmarking of original algorithms GWO and jSO is highlighted in bold. optimization algorithms on a set of 30 problems with From the results, it can be observed that as the stagnareal function. They provide us an opportunity to test the tion parameter increases, the number of cases where efectiveness of our optimization algorithms on a set of the Cooperation algorithm brings improvement also inproblems with real function [7]. creases, initially significantly, but only up to a certain In this section, we discuss the results of the presented Co- value, specifically = 90. Increasing it to = 120 no operation algorithm, which was tested on the CEC2014 longer brings improvement. In fact, there is a little degrafunction set, and its performance was compared with dation. This demonstrates that stagnation is a signifithe original GWO and jSO algorithms. We will examine cant parameter for achieving improvement and should how these two algorithms solved the optimization prob- be chosen carefully. Experiments involving dynamic adlems and compare their results to evaluate the eficiency justment of the stagnation parameter have even greater of the new algorithm. The results provide insights into potential for achieving better results than trying to find the strengths and weaknesses of these optimization algo- the optimal stagnation setting. We plan to focus on this rithms in direct comparison. aspect in the future.
The CEC2014 functions represent the search space for The most interesting fact, as shown by the results of this experiment, is that even though jSO outperforms GWO in only the Cooperation algorithm does fail to bring signifiall cases, their Cooperation brings a noticeable improve- cant improvement, but in most cases, the results are even ment. The GWO algorithm supports faster convergence, worse compared to the original algorithms in both tested while jSO contributes to finding a more accurate opti- dimensions, = 10 and = 30. mum. When setting the stagnation parameter to = 60, the situation noticeably improves, but the results are still 4.2. Test results in dimension = 30 worse for more than half of the tested functions in both dimensions. In contrast, at stagnation parameter = 90, The results of testing the algorithms in dimension = the results become quite satisfactory, with more than 30 on all CEC2014 functions are shown in Table 2. The one-third of the cases outperforming the original funcstructure of the table is the same as described in the sec- tions, and less than one-third of the cases yielding worse tion for summarizing the results in dimension = 10. results. The parameter = 120 does not bring any furAnd again, the medians are displayed and compared. ther significant improvement, but the results do not difer In contrast to the = 10 dimension, GWO yields bet- much from = 90 and are still more than satisfactory. ter results in the = 30 dimension. Out of a total of Recall that all compared values are medians of 15 algo30 functions, GWO outperforms jSO in 3 cases (median rithm runs. As we can see, the stagnation parameter of results is lower). While this is still not a significant significantly afects the performance of the Cooperation number, it suggests that in higher dimensions ( = 50, algorithm, and the best setting is somewhere between = 100, etc.), GWO may still be more beneficial. We = 90 and = 120. This is the same in both tested plan to verify this hypothesis in future research. dimensions = 10 and = 30. The results obtained in the = 30 dimension provide We believe that the reasons for this behavior are as folfurther confirmation that increasing the stagnation pa- lows. When the stagnation parameter is set low ( = 30), rameter leads to significant improvements in Coopera- the algorithms switch too early. The algorithm switch oction, particularly in the initial stages (improvement be- curs before the algorithm can search a suficient number tween = 30 and = 60 is significant), however, in of points and "focus" on the area of search space closest = 90 and = 120, the improvements become either to the current optimum. In contrast, when the stagnation marginal or non-existent. Compared to the results in parameter is set high ( = 120), the algorithm runs too = 10, there is not as much improvement in = 30, long when it is no longer improving, wasting time for but the general trend of improvement is the same. the next algorithm that could have already been run. In future, we expect to achieve even better results by 4.3. Summary of results in dimensions using an adaptive parameter dependent on the computa = 10 and = 30 tion runtime separately for GWO and jSO. This is how we plan to get the most eficiency out of both algorithms.
In the first table section, we can observe the outcomes 5. Conclusion of the original GWO and jSO algorithms, indicating the number of functions for which they provided better re- In this paper, we investigated an algorithm we develsults. We can compare the results in both = 10 and oped, named Cooperation, which combines two well = 30 and the better value is highlighted in bold. established and successful optimization algorithms, GWO The subsequent table section displays the frequencies and jSO. Each of these algorithms is executed for a peat which the Cooperation algorithm, with varying stag- riod of time, determined by a stagnation state, where no nation settings, outperformed the original algorithms. further improvement is observed. We tested diferent Again, we can compare the results in both = 10 and settings of the stagnation parameter and identified its = 30 and the better value is highlighted in bold. significant impact on the overall performance of the new The next table section displays the frequencies at which algorithm, independent of the problem dimension. The the Cooperation algorithm, with varying stagnation set- results are promising because we found a suitable setting tings, provided worse results than the better of the two for the stagnation parameter. However, our future plans original algorithms. include achieving even better results by further adjusting And the last table section illustrates the instances in not only the stagnation parameter. which the Cooperation algorithm, with various stagnation settings, yielded the same result as the better of the References two original algorithms.
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Median comparison: Number of wins Number of wins Number of times Coop 30 is better than the winner Number of times Coop 60 is better than the winner Number of times Coop 90 is better than the winner Number of times Coop 120 is better than the winner Number of times Coop 30 is worse than the winner Number of times Coop 60 is worse than the winner Number of times Coop 90 is worse than the winner Number of times Coop 120 is worse than the winner Number of times Coop 30 is same as the winner Number of times Coop 60 is same as the winner Number of times Coop 90 is same as the winner Number of times Coop 120 is same as the winner 30 0 27 3 1 7 7 8 21 15 15 13