OPTIMIZATION METHOD BASED ON THE SYNTHESIS OF CLONAL SELECTION AND ANNEALING SIMULATION ALGORITHMS

Context. The problem of increasing the efficiency of optimization methods by synthesizing metaheuristics is considered. The object of the research is the process of finding a solution to optimization problems. Objective. The goal of the work is to increase the efficiency of searching for a quasi-optimal solution at the expense of a metaheuristic method based on the synthesis of clonal selection and annealing simulation algorithms. Method. The proposed optimization method improves the clonal selection algorithm by dynamically changing based on the annealing simulation algorithm of the mutation step, the mutation probability, the number of potential solutions to be replaced. This reduces the risk of hitting the local optimum through extensive exploration of the search space at the initial iterations and guarantees convergence due to the focus of the search at the final iterations. The proposed optimization method makes it possible to find a conditional minimum through a dynamic penalty function, the value of which increases with increasing iteration number. The proposed optimization method admits non-binary potential solutions in the mutation operator by using the standard normal distribution instead of the uniform distribution. Results. The proposed optimization method was programmatically implemented using the CUDA parallel processing technology and studied for the problem of finding the conditional minimum of a function, the optimal separation problem of a discrete set, the traveling salesman problem, the backpack problem on their corresponding problem-oriented databases. The results obtained allowed to investigate the dependence of the parameter values on the probability of mutation. Conclusions. The conducted experiments have confirmed the performance of the proposed method and allow us to recommend it for use in practice in solving optimization problems. Prospects for further research are to create intelligent parallel and distributed computer systems for general and special purposes, which use the proposed method for problems of numerical and combinatorial optimization, machine learning and pattern recognition, forecast.

x is a solution (antibody); * x is a (quasi) optimal solution (antibody); ) (⋅ F is the fitness function; ) (⋅ f is the function for which the conditional minimum should be found; is the penalty function; ) (⋅ Φ is an affinity; is a population at iteration n ; μ is population and intermediate population power; C is a set of antibody clones; q is a number of clones for each antibody; C is a set of mutated antibodies clones; λ is a power of the set mutated clones; δ is the mutation parameter, 1 0 < δ < ; ) (n T is the annealing temperature at iteration n ; 0 T is an initial annealing temperature; β is a cooling ratio; is the z th equality constraint; ) (⋅ z g is the z th inequality constraint; 1 Z is a number of equality constraints; 2 Z is a number of inequality constraints; w X is the subset of the current population containing antibodies that satisfy all constraints; 1 I is a number of arguments of the function for which the conditional minimum should be found; S is the partitionable set; k s is a vector, one-to-one corresponding to the k th element of the partitionable set; J is a number of features of the element partitionable set; K is the power of the partitionable set; 2 I is a number of classes into the partitionable set;

INTRODUCTION
Today, the development of methods aimed at solving problems of numerical and combinatorial optimization, machine learning, etc., which are used in general and special-purpose intelligent computer systems, is an urgent task.
Existing optimization methods that find the exact solution have high computational complexity. Optimization and machine learning methods that find an approximate solution through directional search have a high probability of falling into a local extremum. Random search methods do not guarantee convergence. In this connection, the problem of insufficient efficiency of optimization methods, which needs to be solved, arises.
The object of study is the process of finding solutions to optimization problems.
The subject of study is the methods for finding a quasi-optimal solution based on metaheuristics.
The purpose of the work is to increase the efficiency of searching for a quasi-optimal solution at the expense of a metaheuristic method based on the synthesis of clonal selection and annealing simulation algorithms.
To achieve this goal, it is necessary to solve the following tasks: 1) to create a quasi-optimal method based on the synthesis of clonal selection and annealing simulation algorithms; 2) to adapt the proposed method to the problem of finding the conditional minimum of functions; 3) to adapt the proposed method to the problem of optimal partitioning of a discrete set; 4) to adapt the proposed method to the traveling salesman problem; 5) to adapt the proposed method to the knapsack problem; 6) to conduct a numerical study of the proposed optimization method.

PROBLEM STATEMENT
The problem of increasing the efficiency of searching a solution to an optimization problem based on clonal selection is represented as the problem of finding such an

REVIEW OF THE LITERATURE
To accelerate a quasi-optimal solution of optimization and machine learning problems and reduce the likelihood of falling into a local extremum, metaheuristics (or advanced heuristics) are used [1][2][3][4][5]. Metaheuristics expands the capabilities of heuristics by combining heuristic methods based on a high-level strategy [6][7][8][9][10].
However, modern metaheuristics have one or more of the following disadvantages: -there is only an abstract description of the method or the description of the method is focused on solving only a specific task [1]; -the influence of the iteration number on the solution search process [2] is not taken into account; -the convergence of the method [11] is not guaranteed; -there is no possibility of using non-binary potential solutions [12]; -the procedure for determining the values of parameters [13] is not automated; -there is no possibility to solve the problems of conditional optimization [14]; -insufficient accuracy of the method [15]. Therefore, the efficiency of the method for the search of quasi-optimal solution is of paramount importance.

MATERIALS AND METHODS
The optimization method based on the synthesis of clonal selection and annealing simulation algorithms is developed.
The sequence of procedures of the proposed optimization method based on the synthesis of clonal selection and annealing simulation algorithms is shown in Fig. 1.

In block 1, an initial population
with power μ is created, and each antibody of this population corresponds to a potential solution of the problem.
In block 2, the current annealing temperature at itera- In block 3, the value of the fitness function ) (x F for each antibody x , which is determined by the specificity of the particular optimization problem, is calculated.
In block 4, the affinity value for each antibody x is calculated.
Affinity is a function that determines the proximity of antibody x to the best antibody in the current population and is calculated based on the utility function. The affinity value is calculated as , then the antibody is the best.
, then the antibody is the worst. In block 5, the mutation probability for each antibody x which depends on the affinity value In block 6, the cloning operator for each antibody x is executed.
The cloning operator cl A plays a role similar to the operator of genetic algorithm reproduction.
The number of clones q for each antibody x is de- As a result of applying the cloning operator cl A to the , a set of antibody clones } {c C = are formed. In block 7, the mutation operator for each clone c is executed.
The mutation operator mt A allows to obtain new antibodies from antibody clones with sharply different properties.
The mutation based on the annealing simulation over each component of each clone c is executed when The features of the proposed variant of the mutation operator are the following: − there is an inverse relationship between the mutation probability and the affinity value, i.e. the best (in terms of affinity) clones change less often than the worst (in terms of affinity) clones; − there is an inverse relationship between the mutation probability and the iteration number, i.e. at the initial iterations the entire search space is explored and at the final iterations the search becomes directional.
As a result of applying the mutation operator mt A to a set of antibody clones } {c C = , a set of mutated is formed. In block 8, the reduction operator is executed.
As the reduction operator rd A , the scheme ) ( λ + μ [9] is used, which provides the direction of the search (the best antibodies are preserved) and In block 9, a dynamic replacement operator is executed.
For a broader study of the search space an operator rp A , which replaces the last (the worst by affinity) antibodies of the intermediate population with new antibodies, is used.
The number of replaced antibodies d is determined as A peculiarity of the proposed dynamic replacement operator is the following -there is an inverse relationship between new antibodies number and the iteration number, i.e. at the initial iterations the entire search space is explored and at the final iterations the search becomes directional.
As a result of the application of the replacement op- In block 10, the best antibody by the value of the fit- is determined.
In block 11, the condition for completing the solution search is checked.
, then a quasi-optimal solution * x is obtained.
The adaptation of the optimization method based on the synthesis of clonal selection and annealing simulation algorithms for the problem of finding the conditional minimum of the function is given.
The proposed method is used to minimize the function, taking into account equality constraints and inequality constraints. For this task, each antibody is a collection of function arguments, and blocks 1, 3, 7, 9 have the following features.
In block 1 and block 9, each component i x of each antibody x is initialized as 1. The creation of initial population ( ) In block 3, the value of the fitness function ) (x F for each antibody x is calculated as In block 7, a mutation based on annealing simulation algorithm over each component i c of each clone c is executed as For this task, the mutation operator, besides the features indicated in the description of the developed method, has the following additional features: -the value of each component i c is always in the al- ; -there is an inverse relationship between the magnitude of the mutation step and the affinity value, i.e. the best (in terms of affinity) clones change less than the worst (in terms of affinity) clones; -there is an inverse relationship between the magnitude of the mutation step and the iteration number, i.e. at the initial iterations the entire search space is explored and at the final iterations the search becomes directional; -it does not require the use of binary potential solutions, i.e. there is no need to convert real potential solutions into binary ones before the mutation and to convert binary potential solutions into real ones after the mutation, which reduces the computational complexity of the mutation operator and speeds up the search for a solution.
The adaptation of the optimization method based on the synthesis of clonal selection and annealing simulation algorithms to the problem of optimal discrete set partitioning is given.
The proposed method is used to minimize the rootmean-square error of the partition of a finite discrete set into a given number of classes. For this task, each antibody is a set of class centers, and blocks 1, 3, 7, 9 have the following features.
In block 1 and block 9, each component ij x of each antibody x is initialized as In block 3, the value of the fitness function ) (x F for each antibody x is calculated as 2 , 1 2 min arg In block 7, a mutation based on the annealing simulation algorithm over each component ij c of each clone c is executed as For this task, the mutation operator, besides the features indicated in the description of the developed method, has the following additional features: -the value of each component ij c is always in the al- -there is an inverse relationship between the magnitude of the mutation step and the affinity value, i.e. the best (in terms of affinity) clones change less than the worst (in terms of affinity) clones; -there is an inverse relationship between the magnitude of the mutation step and the iteration number, i.e. at the initial iterations the entire search space is explored and at the final iterations the search becomes directional; -it does not require the use of binary potential solutions, i.e. there is no need to convert real potential solutions into binary ones before the mutation and convert binary potential solutions into real ones after the mutation, which reduces the computational complexity of the mutation operator and speeds up the search for a solution.
The adaptation of the optimization method based on the synthesis of clonal selection and annealing simulation algorithms for the traveling salesman problem is given.
The proposed method is used to minimize the length of the route, passing only once through all points. For this task, each antibody is a collection of route points, and blocks 1, 3, 7, 9 have the following features.
In block 1 and block 9, each component i x of each antibody x is initialized with a randomly selected route point number, and the point numbers should not be duplicated.
In block 3, the value of the fitness function ) (x F for each antibody x is calculated as In block 7, the mutation based on an annealing simulation algorithm over each component i c and a randomly selected component k c of each clone is executed as For this task, the mutation operator, besides the features indicated in the description of the developed method, has the following additional features: -the value of each component always belongs to an admissible set } ,..., 1 { 3 I ; -it does not require the use of binary potential solutions, i.e. there is no need to convert integer potential solutions into binary ones before the mutation and to convert binary potential solutions into real ones after mutation, which reduces the computational complexity of the mutation operator and speeds up the search for a solution.
The adaptation of the optimization method based on the synthesis of clonal selection and annealing simulation algorithms for the knapsack problem is given.
The proposed method is used to select from a given set of objects with the properties "cost" and "weight" the subset with the maximum cost, while observing the limit on the total weight. For this task, each antibody is a collection of weights, and blocks 1, 3, 7, 9 have the following features.
In block 1 and block 9, each component i x of each antibody x is initialized as In block 3, the value of the fitness function ) (x F for each antibody x is calculated as In block 7, the mutation based on an annealing simulation algorithm over each component i c of each clone c is executed as For this problem, the mutation operator has the features indicated in the description of the developed method.

EXPERIMENTS
A numerical study of the proposed optimization method was carried out using the CUDA technology of information parallel processing, the number of threads in the block corresponded to the population size, the population was sorted based on the paired-disparity sorting algorithm, the antibody with the lowest value of the fitness function was searched. Let , and, moreover, x ; -optimal partitioning of a discrete set, the search for a solution was carried out on the standard BSDS500 database; -"the traveling salesman", the search for a solution was conducted on the standard berlin52 database; -"the knapsack", the search for a solution was carried out on the standard KNAPSACK_01 database.
The study leads to the conclusion that the proposed method provides a high accuracy of finding a solution.

RESULTS
The function of the annealing temperature decrease is determined by the formula 0 ) ( T n T n β = and is shown in Fig. 2. The dependence (Fig. 2) of the annealing temperature on the iteration number shows that the annealing temperature decreases with increasing of the iteration number.
The mutation probability in the case for the worst an- and is shown in Fig. 3. The dependence (Fig. 3) of the mutation probability on the annealing temperature shows that the mutation probability decreases with temperature decreasing.
The results of the comparison of the proposed method with the method based on the theory of clonal selection and described in [16][17][18][19][20][21][22] are presented in Table 1.

DISCUSSION
The selected values of the parameters of the proposed optimization method provide a high probability of mutation at the initial iterations and a low probability of mutation at the final iterations. For example, for the worst antibody with the maximum probability of mutation, the mutation occurs with a probability of no less than 0.9 for the first 40% iterations and with a probability below 0.1 for the last 10% iterations (Fig. 3).
Method based on clonal selection theory [16][17][18][19][20][21][22]: -does not take into account the iteration number in the operator of mutation and replacement, which reduces the accuracy of the search for a solution (Table 1); -does not allow real potential solutions in the mutation operator, which increases the computational complexity of the mutation operator and slows down the search for a solution. This is due to the need to convert non-binary potential solutions into binary before the mutation and binary potential solutions into non-binary after mutation (Table 1); -does not allow to find the conditional extremum. The proposed method allows to eliminate these drawbacks.

CONCLUSIONS
In this paper, the actual scientific and technical problem of increasing the efficiency of optimization methods was solved by dint of creates the method of finding a quasi-optimal solution by a metaheuristic.
The scientific novelty of obtained results is that the optimization method based on the synthesis of clonal selection and annealing simulation algorithms is proposed. It allows to increase the search accuracy through the application of the principle of organizing the study of the entire search space at the initial iterations and focusing of the search on the final iterations.
The adaptation of the proposed method both for the problem of finding the conditional minimum of functions and for the problem of optimal partitioning of a discrete set: -allows real potential solutions in the mutation operator, which reduces the computational complexity of the mutation operator and speeds up the search for a solution; -uses a dynamic mutation step, which allows to investigate the entire search space at the initial iterations and to make the search directional at the final iterations, that ensures high accuracy of the search.
The solution of finding the problem of conditional minimum of functions and the knapsack problem by the proposed method uses a penalty function, which allows to find a conditional extremum.
In addition, the application of the proposed method for solving the traveling salesman problem allows integer potential solutions in the mutation operator, which reduces the computational complexity of the mutation operator and speeds up the search for a solution.
The practical significance of the obtained results lies in the fact that the scope of application of metaheuristics is expanding on the basis of the theory of clonal selection by adapting the proposed method for the indicated optimization problems. This contributes to the effectiveness of intelligent computer systems for general and special purposes.
Prospects for further research are the study of the proposed method for a wide class of artificial intelligence tasks.

ACKNOWLEDGEMENTS
The studies were carried out in accordance with the priority direction of the development of science and technology in Ukraine until 2020 "Information and Communication Technologies" and contain some results of the state budget scientific research project "Methods, models for the processing of intelligent, information technologies for highly efficient computing and local control subsystems in problem-oriented systems" (state registration number 0106U004501) and "Basic components of microprocessor control systems by laser technological complexes on the basis of table-algorithmic methods, models and incomplete similarity theory" (state registration number 0113U003345).