Search Optimization Opportunities of Modified Self-Organizing Migrating Algorithm in Multi-Extremal Tasks Environment

The paper studies are the search optimization task of multi-extremal objects, which are more complicated than mono-extremal. Paper postulates that to nd extreme suitable values on complex test function the heuristic algorithm is one way. Self-Organizing Migrating Algorithm and devised approach applied to this task are considered. Conducted research established common test environment to compare multiextremal test functions. Speci c characteristics for problem solving of detection and identi cation of global and local extreme are included. Additional clustering mechanism is described. Obtained measurements of Self-Organizing Migrating Algorithm on a range of multi-extremal test functions are illustrated.


Introduction
The most advanced problems in science, technology, economics, military aairs and other applied modern trends are connected with the tasks of nding optimums in designs, technologies, models and environments, through the possibility of controlling the dynamic and static states, as well as, other requirements put forward in the specications of the design objects.
In other words, the developers have to solve the problems of Searching Optimization (SO) [1]. It is very typical that most of the current known SO methods are developed and eectively used to nd only one extreme, which is often the global one.
However, many tasks in solving complex technological systems and transportation problems require optimization. Especially, the objects of discrete nature are characterized by MultiExtremal (ME) properties [2] and [3]. A signicant distinctive property for solving such tasks requires specic methods to reach the solution. It is unlikely that these methods should be sought in the class of the SO deterministic methods, though such attempts are already well known.
These methods are too sensitive to the sign variation of discontinuous functions within their continuum response factor spaces. For solving real optimization problems, it has been common to apply methods called heuristic. These methods are the most perspective to obtain solutions for the ME problems [4] and [5]. The Self-Organizing Migrating Algorithm (SOMA) [6], [7] and [8]. SOMA is an algorithm developed in 1999, whose operation is based, like the Scatter Search [9] or Particle Swarm Optimization (PSO) [10] on vector operations.
The original idea that led to its creation, is to mimic the behavior of a group of intelligent individuals who cooperate in solving common problems such as nding food sources, etc.
Since working with similar populations, such as Genetic Algorithms (GA) [11] and evolutionary outcome after one it is identical with genetic algorithm and dierential evolution, it can be understood like sort of evolutionary algorithm despite the fact that during its run it is not in terms of philosophy of creating a new algorithm descendants, as in other classical evolutionary algorithms.
To test the eectiveness of the developed modications require Test Functions (TFs) with not only global extremum, but also a variety of global, local or sub-extreme values. The authors are chosen next TFs: Rastrigin [12], Rosenbrock [13], Himmelblau [14], Lambda [15], Schwefel [16], Giunta [17], Ursem [18], Shubert [19] and • All to All Adaptive. It is a modied All to All strategy. Individuals do not begin a new migration from the same old position, but from the last best position found during the last traveling to the previous individual.
As a result of previously conducted research [6] the most logical strategy is "All to All Adaptive", due to the high eciency (in spite of the increasing computing time). Also, the authors attempt was create new strategies ("All to One plus Random", "All to Neighbor and All to One plus Neighbor"), but the results of computational experiments have shown that the basic strategies were more eective.

SOMA Principles
The canonical SOMA version [6] consists of the 3. Modied SOMA for ME SO Tasks ME SOMA modication based on the canonical SOMA, but it has additional discrete mechanism and clustering process. After allocating all extreme in each space the result values are clustered, to get a true picture without "nearly-extremum" values.
Clustering does not require resulting number of clusters. It can be described by the following equations: The matrix of normal similarity measures: where: x is the plurality of elements; Q is a number of elements in plurality; q, i ∈ 1 . . . Q, d(x, y) is a clustering criterion (like Euclidean distance between points, etc.).
The matrix of similarity measures of elements plurality: where a, b ∈ x plurality.
The result matrix: where R ζ is relation between clustering points.
Values in result R matrix will show whether the pair of points belongs R relation, their called "quasi-equivalence levels" (a). The choice of a particular level divides the plurality into equivalence classes, which correspond to separate clusters. Fig. 1

demonstrates ow-chart of "A Quasi
Equivalence" clustering algorithm. ME SOMA modication requires to do "A Quasi-Equivalence" clustering by 2 dierent criterion: rst -by Euclidean distance between

Check ME SOMA Eectivness
To test the eectiveness of developed ME SOMA modication chosen 9 famous TF. An appropriate software tool "ME SOMA" was also devel- N is dependent on TF, M igration = 10, P opSize = 7, Step = 0.11, P athLength = 3, P RT = 0.1, M inDiv = 1e −15 . Strategy is "All to All Adaptive". a is dependent on TF To demonstrate ME SOMA modication result the gures for each TF with allocated extreme area clusters are illustrated. As can seen   The experiments described above have shown that to find the global and local extrema at ME TF recommended to use N>5. Also, a parameter can significantly improve the drop-out process of sub-local populations. To optimize TF with plurality of global extrema recommended use a<0.85. To optimize TF with plurality of global and local extrema recommended use a>0.85.
For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.
V. COMPARING ME SOMA RESULT ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3    For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.
V. COMPARING ME SOMA RESULT ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3    For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.
V. COMPARING ME SOMA RESULT ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3 Fig. 9. Shubert TF. N=4, a=0.95. Fig. 10. Plateau TF. N=5, a=0.95.
The experiments described above have shown that to find the global and local extrema at ME TF recommended to use N>5. Also, a parameter can significantly improve the drop-out process of sub-local populations. To optimize TF with plurality of global extrema recommended use a<0.85. To optimize TF with plurality of global and local extrema recommended use a>0.85.
For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.

V. COMPARING ME SOMA RESULT
ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3 Fig. 9. Shubert TF. N=4, a=0.95. Fig. 10. Plateau TF. N=5, a=0.95.
The experiments described above have shown that to find the global and local extrema at ME TF recommended to use N>5. Also, a parameter can significantly improve the drop-out process of sub-local populations. To optimize TF with plurality of global extrema recommended use a<0.85. To optimize TF with plurality of global and local extrema recommended use a>0.85.
For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.

V. COMPARING ME SOMA RESULT
ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3 Fig. 9. Shubert TF. N=4, a=0.95. Fig. 10. Plateau TF. N=5, a=0.95.
The experiments described above have shown that to find the global and local extrema at ME TF recommended to use N>5. Also, a parameter can significantly improve the drop-out process of sub-local populations. To optimize TF with plurality of global extrema recommended use a<0.85. To optimize TF with plurality of global and local extrema recommended use a>0.85.
For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.

V. COMPARING ME SOMA RESULT
ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3 demonstrates the (f) Giunta TF. N = 4, a = 0.85.    Fig. 9. Shubert TF. N=4, a=0.95. Fig. 10. Plateau TF. N=5, a=0.95.
The experiments described above have shown that to find the global and local extrema at ME TF recommended to use N>5. Also, a parameter can significantly improve the drop-out process of sub-local populations. To optimize TF with plurality of global extrema recommended use a<0.85. To optimize TF with plurality of global and local extrema recommended use a>0.85.
For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.

V. COMPARING ME SOMA RESULT
ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3 Fig. 6. Schwefel TF. N=5, a=0.85. Fig. 7. Giunta TF. N=4, a=0.85. Fig. 8. Ursem TF. N=4, a=0.85. Fig. 9. Shubert TF. N=4, a=0.95. Fig. 10. Plateau TF. N=5, a=0.95.
The experiments described above have shown that to find the global and local extrema at ME TF recommended to use N>5. Also, a parameter can significantly improve the drop-out process of sub-local populations. To optimize TF with plurality of global extrema recommended use a<0.85. To optimize TF with plurality of global and local extrema recommended use a>0.85.
For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.

V. COMPARING ME SOMA RESULT
ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3    The experiments described above have shown that to find the global and local extrema at ME TF recommended to use N>5. Also, a parameter can significantly improve the drop-out process of sub-local populations. To optimize TF with plurality of global extrema recommended use a<0.85. To optimize TF with plurality of global and local extrema recommended use a>0.85.
For the best result of modification on different ME TFs, additional sub-optimization of the method parameters is necessary.

V. COMPARING ME SOMA RESULT
ME SOMA comparing with other analogues has two problems: firstis a little number of ME SO algorithm and secondis papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modifications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme. Table 3 demonstrates the (i) Plateau TF. N = 5, a = 0.95.

Comparing ME SOMA Result
ME SOMA comparing with other analogues has two problems: rst -is a little number of ME SO algorithm and second -is papers, which describing these algorithms usually do not contain any numerical experimental data or optimization problem is far from TF. Therefore, the comparison will be with the results presented in the [23], which described modications of PSO, GA and Ant Colony Optimization (ACO). Table 2 illustrates standard values of Rastrigin TF extreme.

Related Work
In the design optimization process, we are often confronted with problems facing the ME conditions. Such situation requires decisions, which take into consideration several identical or close extremes, and the best choice in-between them has to be made. The classical theory of schedul- ing gives examples, where several identical op-timums and identical sub-optimums, close to them exist [1], [3], [4] and [5]. The majority of discrete, integer and combinatory programming problems diers in such property [24], [22], [23], [24] and [25], in particular, when nding solution for graphs [26], [27], [28] and [29]. Thenite number of admissible decisions requires considering the ME solutions for the discrete environment optimization. There are many additional conditions, which can help to choose the extreme, equivalent or close in size, and satisfy both, the numerical criteria estimates and the heuristic ideas. Therefore, the choice, of the most eective methods and algorithms, is an extremely important step to nd such solution of the ME task.

Conclusions
The analysis of SOMA application for solving the ME tasks showed that modication is efcient, eective, and bring some essential features to the presented solutions. The specic approaches to solve the task for each of these particular cases is determined through the analysis of the algorithm features; the detection and identication of local extremes, clustering method and subsequent operations resulting from such analysis. Also, ME SOMA modication showed reasonable performance.
To conclude, studied SOMA is relevant and promising for future applications. The specic choice of the algorithm tool for solving ME tasks depends on the experience and personal researcher preferences, as well as on the special features of the domain specic research area.