CN116070672A - Optimization problem solving method based on improved whale optimization algorithm - Google Patents

Optimization problem solving method based on improved whale optimization algorithm Download PDF

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CN116070672A
CN116070672A CN202310080915.7A CN202310080915A CN116070672A CN 116070672 A CN116070672 A CN 116070672A CN 202310080915 A CN202310080915 A CN 202310080915A CN 116070672 A CN116070672 A CN 116070672A
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许志猛
李丹
陈志璋
陈良琴
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Abstract

The invention relates to an optimization problem solving method based on an improved whale optimization algorithm, which comprises the following steps: based on the whale optimization algorithm, an improved whale optimization algorithm based on quasi-reflection learning is constructed, and the method comprises the following steps: 1) Initializing a population based on quasi-reflection learning; 2) Optimizing whale individuals in the population by adopting nonlinear convergence factors, and updating the individual positions; 3) Adopting a random individual-based position updating strategy to secondarily update the positions of the population individuals; and solving the engineering field optimization problem by using the obtained improved whale optimization algorithm. The method is beneficial to improving the precision and speed of optimization.

Description

Optimization problem solving method based on improved whale optimization algorithm
Technical Field
The invention relates to the field of engineering design planning, in particular to an optimization problem solving method based on an improved whale optimization algorithm.
Background
The core problem of many projects is ultimately ascribed to optimization problems. Therefore, optimization has become an indispensable computing tool for engineering technicians. The optimization problem is a problem of determining what values should be taken by some selectable variables under certain constraint conditions to optimize a selected objective function, and is widely applied to the fields of production management, economic planning, engineering design, system control and the like. Considering that the problems to be solved in the real world generally have the characteristics of diversity and complexity, in order to accurately solve the optimization problems, the number of variables or the number of decision variable dimensions in the optimization scheme needs to be continuously increased, which causes the problems of large calculation amount, dimension disaster and the like in the solution of the optimization problems. Therefore, the intelligent optimization algorithm is widely applied due to the characteristics of simple concept, easiness in implementation, no gradient information, avoidance of local optimal solution and the like. The intelligent optimization algorithm is a random search algorithm based on biological intelligence or natural phenomena, and the main idea is to simulate the behavior of some social species in nature to form a mathematical model for solving various problems. The algorithm can search in a global way to a certain extent, find an approximate solution of the optimal solution, and has important application value for solving the optimal solution or satisfactory solution of the complex optimization problem.
The whale optimization algorithm is a biological population intelligent algorithm, and the algorithm is a novel biological population intelligent algorithm which is proposed by Australian scholars Mirjallii and Lewis in 2016 according to unique bubble network foraging behaviors of whales at the head. As a novel biological population intelligent algorithm, the algorithm has good effect on convergence accuracy and convergence speed, but in the optimization solution of the actual problem, the condition that the solution accuracy is poor and the iteration process is easy to fall into local optimum still exists. For these problems, researchers have combined the whale optimization algorithm with differential evolution, and some researchers have proposed whale optimization algorithms based on opposite learning and exponential optimization, and in addition, chaos theory is introduced into the optimization process of whale optimization algorithms. However, these whale optimization algorithm improvements have not completely solved the problem of unbalanced global searching capability and local development capability, and of being prone to local optimality. Therefore, there is a need for improvements to existing whale optimization algorithms to increase the optimization effect based on whale optimization algorithms.
Disclosure of Invention
The invention aims to provide an optimization problem solving method based on an improved whale optimization algorithm, which is beneficial to improving the precision and speed of optimization.
In order to achieve the above purpose, the invention adopts the following technical scheme: an optimization problem solving method based on an improved whale optimization algorithm comprises the following steps:
based on the whale optimization algorithm, an improved whale optimization algorithm based on quasi-reflection learning is constructed, and the method comprises the following steps:
1) Initializing a population based on quasi-reflection learning;
2) Optimizing whale individuals in the population by adopting nonlinear convergence factors, and updating the individual positions;
3) Adopting a random individual-based position updating strategy to secondarily update the positions of the population individuals;
and solving the engineering field optimization problem by using the obtained improved whale optimization algorithm.
Further, in step 1), the implementation method for initializing the population based on quasi-reflection learning is as follows:
Figure BDA0004067406890000021
wherein ,
Figure BDA0004067406890000022
representing the position of the i-th whale in the j-dimensional space,/I>
Figure BDA0004067406890000023
and />
Figure BDA0004067406890000024
Respectively indicate->
Figure BDA0004067406890000025
And rand (0, 1) is a random number between 0 and 1.
Further, in the step 2), the implementation method for optimizing whale individuals in the population by adopting the nonlinear convergence factor comprises the following steps:
the optimizing process comprises three optimizing modes, namely surrounding a prey, foaming net attack and searching predation;
surrounding the prey: suppose that in d-dimensional space, the current best whale individual X * The position of (2) is
Figure BDA0004067406890000026
Whale individual X j The position of (2) is +.>
Figure BDA0004067406890000027
The next position X of the whale individual under the influence of the optimal position j+1 Is that
Figure BDA0004067406890000028
The specific mathematical model is as follows:
Figure BDA0004067406890000029
Figure BDA00040674068900000210
C 1 ＝2r 2
A 1 ＝2a·r 1 -a
Figure BDA00040674068900000211
wherein ,
Figure BDA00040674068900000212
representing the space coordinate X j+1 The kth component, r 1 and r2 Are all random numbers between 0 and 1, max iter The maximum iteration number is t, and the current iteration number is t;
foaming net attack: two mathematical models were constructed to simulate this predation behavior, respectively:
a) Shrink wrapping; the mathematical model constructed by the predation behavior is almost identical with the mathematical model surrounding the prey, and one difference is A 1 Is a value range of (a); since the contracted envelope is to bring the whale individual at the current position close to the whale individual at the current optimal position, A 1 The value of (a) is in the range from original [ -a, a]Adjust to [ -1,1]Other formulas remain unchanged;
b) Spiral position updating; the predation behavior is that the current whale individual approaches to the current optimal whale individual in a spiral mode, and a specific mathematical model is as follows:
Figure BDA0004067406890000031
Figure BDA0004067406890000032
wherein b is a logarithmic spiral shape constant, and l is a random number between-1 and 1;
the whale in the seat not only shrinks the wrapping ring, but also walks in a spiral form towards the prey, so that the position update chooses either to shrink the wrapping ring or to walk in a spiral form towards the prey with 50% probability, respectively, the specific mathematical model is as follows:
Figure BDA0004067406890000033
searching for predation: in a mathematical model of shrinkage surrounding this predatory behaviour A 1 Is of the value of (2)The range is limited to [ -1,1]But when A 1 The value of (2) is not within [ -1,1]At this time, the current whale individual may not approach the current best whale individual, but randomly select one whale individual from the current whale population; suppose that in d-dimensional space, one whale individual X is randomized in the current whale population rand The position of (2) is
Figure BDA0004067406890000034
Whale individual X j The position of (2) is +.>
Figure BDA0004067406890000035
The mathematical model of search predation behavior is as follows:
Figure BDA0004067406890000036
Figure BDA0004067406890000037
C 1 ＝2r 2
A 1 ＝2a·r 1 -a
Figure BDA0004067406890000038
further, in step 3), the implementation method of the location update policy based on the random individual is as follows:
Figure BDA0004067406890000039
each whale individual goes through the phases of surrounding predation, spiral update and hunting, after which each individual updates its position again by a position update strategy based on random individuals, taking the optimal position before and after the update.
Further, the method is applied to channel estimation value optimization in channel estimation; firstCombining LS algorithm and MMSE algorithm to construct a channel estimation value
Figure BDA0004067406890000041
There are two unknown parameters x in the constructed channel estimate 1 、x 2 To characterize the transmitted signal; to obtain a more accurate channel estimate, a modified whale optimization algorithm is used to select the appropriate parameter x 1 、x 2 To minimize the mean square error MSE of the estimated and actual channels; the optimization model for channel estimation is as follows:
Figure BDA0004067406890000042
Figure BDA0004067406890000043
Figure BDA0004067406890000044
Figure BDA0004067406890000045
-1＜x 1 ＜1，-1＜x 2 ＜1
wherein ,
Figure BDA0004067406890000046
representing the channel estimate obtained by improving the whale optimization algorithm, < >>
Figure BDA0004067406890000047
For channel estimation using least squares algorithm LS +.>
Figure BDA0004067406890000048
H is the true channel for channel estimation using minimum mean square error algorithm MMSE, ++>
Figure BDA0004067406890000049
Is a cross-correlation matrix, R HH Is the autocorrelation matrix of the real channel; i is an identity matrix, X is a transmission signal matrix, Y is a reception signal vector, +.>
Figure BDA00040674068900000410
For the power of the transmitted signal, +.>
Figure BDA00040674068900000411
Is the power of the noise signal.
Further, the method is applied to the design of the compression spring; modeling the design of a compression spring as an optimization problem with the design objective of minimizing its mass f (x) under certain constraints, including minimum deflection, strong shear stress, oscillation frequency, and outer diameter limitations of 4 inequality constraints, the average diameter of the spring coil D (x 1 ) Diameter of spring wire d (x) 2 ) The number of effective turns of the spring N (x 3 ) 3 design variables; the correlation model is as follows:
Figure BDA0004067406890000051
Figure BDA0004067406890000052
Figure BDA0004067406890000053
Figure BDA0004067406890000054
Figure BDA0004067406890000055
Figure BDA0004067406890000056
Variable range 0.05≤x 1 ≤2.00
0.25≤x 2 ≤1.30
2.00≤x 3 ≤15.0
solving to obtain the optimal spring design parameters meeting the requirements by improving a whale optimization algorithm: average diameter of spring coil D (x) 1 ) Diameter of spring wire d (x) 2 ) And the effective number of turns of the spring N (x 3 )。
Compared with the prior art, the invention has the following beneficial effects: the method constructs an improved whale optimization algorithm based on quasi-reflection learning, solves the optimization problem in the engineering field through the obtained improved whale optimization algorithm, avoids the problems that the traditional biological population intelligent optimization algorithm is easy to be trapped in local optimization and low in convergence precision, improves the knowledge precision and the convergence speed, and can be widely applied to the fields of solving the optimization problem in engineering design.
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FIG. 1 is a flow chart of a method implementation of an embodiment of the present invention.
Fig. 2 is a plot of MSE convergence obtained over 30 iterations in an embodiment of the invention.
Fig. 3 is a schematic diagram of the application of the method to a compression spring in an embodiment of the invention.
Detailed Description
The invention will be further described with reference to the accompanying drawings and examples.
It should be noted that the following detailed description is exemplary and is intended to provide further explanation of the present application. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments in accordance with the present application. As used herein, the singular is also intended to include the plural unless the context clearly indicates otherwise, and furthermore, it is to be understood that the terms "comprises" and/or "comprising" when used in this specification are taken to specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof.
As shown in fig. 1, the present embodiment provides an optimization problem solving method based on an improved whale optimization algorithm, including:
based on the whale optimization algorithm, an improved whale optimization algorithm based on quasi-reflection learning is constructed, and the method comprises the following steps:
1) Population initialization based on quasi-reflection learning is performed.
2) Optimizing whale individuals in the population by using nonlinear convergence factors, and updating the individual positions.
3) And adopting a random individual-based position updating strategy to secondarily update the positions of the population individuals.
And solving the engineering field optimization problem by using the obtained improved whale optimization algorithm.
1. Population initialization
The high-quality initialized population has very important influence on the solving precision of an optimization algorithm. According to whether the population initialization has randomness, the initialization methods are divided into two main categories: random algorithms and deterministic algorithms. Most of the current optimization algorithms use the simplest pseudo-random number generator to initialize a population. But are not sufficiently uniform in population, especially in high-dimensional space, generated by a pseudo-random number generator. In some specific problems, the population generated by the pseudo-random number generator does not allow the algorithm to achieve a good solution. The invention provides a quasi-reflection learning-based population initialization for improving the quality of an initialized population, which comprises the following implementation steps:
Figure BDA0004067406890000061
wherein ,
Figure BDA0004067406890000062
representing the position of the i-th whale in the j-dimensional space,/I>
Figure BDA0004067406890000063
and />
Figure BDA0004067406890000064
Respectively indicate->
Figure BDA0004067406890000065
And rand (0, 1) is a random number between 0 and 1.
2. Iterative optimization
The iterative optimization is the most core place of the whole optimization algorithm, and the iterative optimization of the biological population intelligent tree algorithm is mainly generated by referring to some activities of living things in the nature. The step size in iterative optimization determines the convergence rate of the algorithm. Most algorithms use a fixed step size for iterative optimization. This causes a situation in which the convergence speed is too slow. The invention provides a method for optimizing whale individuals in a population by adopting nonlinear convergence factors to improve optimizing performance, which comprises the following steps:
the process of optimizing includes three optimizing modes, namely, surrounding the prey, foaming net attack and searching predation.
1. Surrounding the prey: suppose that in d-dimensional space, the current best whale individual X * The position of (2) is
Figure BDA0004067406890000066
Whale individual X j The position of (2) is +.>
Figure BDA0004067406890000067
The next position X of the whale individual under the influence of the optimal position j+1 Is that
Figure BDA0004067406890000068
The specific mathematical model is as follows:
Figure BDA0004067406890000069
Figure BDA0004067406890000071
C 1 ＝2r 2
A 1 ＝2a·r 1 -a
Figure BDA0004067406890000072
/>
wherein ,
Figure BDA0004067406890000073
representing the space coordinate X j+1 The kth component, r 1 and r2 Are all random numbers between 0 and 1, max iter And t is the current iteration number, and is the maximum iteration number.
2. Foaming net attack: two mathematical models were constructed separately to simulate this predation behavior.
a) Shrink wrapping; the mathematical model constructed by the predation behavior is almost identical with the mathematical model surrounding the prey, and one difference is A 1 Is a value range of (a); since the contracted envelope is to bring the whale individual at the current position close to the whale individual at the current optimal position, A 1 The value of (a) is in the range from original [ -a, a]Adjust to [ -1,1]The other formulas remain unchanged.
b) Spiral position updating; the predation behavior is that the current whale individual approaches to the current optimal whale individual in a spiral mode, and a specific mathematical model is as follows:
Figure BDA0004067406890000074
Figure BDA0004067406890000075
wherein b is a logarithmic spiral shape constant, and l is a random number between-1 and 1.
The whale in the seat not only shrinks the wrapping ring, but also walks in a spiral form towards the prey, so that the position update chooses either to shrink the wrapping ring or to walk in a spiral form towards the prey with 50% probability, respectively, the specific mathematical model is as follows:
Figure BDA0004067406890000076
3. searching for predation: in a mathematical model of shrinkage surrounding this predatory behaviour A 1 The range of the value of (C) is limited to [ -1,1]But when A 1 The value of (2) is not within [ -1,1]At this time, the current whale individual may not approach the current best whale individual, but randomly select one whale individual from the current whale population; suppose that in d-dimensional space, one whale individual X is randomized in the current whale population rand The position of (2) is
Figure BDA0004067406890000077
Whale individual X j The position of (2) is +.>
Figure BDA0004067406890000078
The mathematical model of search predation behavior is as follows:
Figure BDA0004067406890000081
Figure BDA0004067406890000082
C 1 ＝2r 2
A 1 ＝2a·r 1 -a
Figure BDA0004067406890000083
3. location update
The location update is that the algorithm updates the current location of each individual in the population after one iterative optimization. A good location update strategy helps the algorithm jump out of local optimization. A common location update is to directly assign the individual the location generated after iterative optimization, which is the simplest and most direct location update strategy. However, this also results in algorithms that tend to fall into local optima. The invention provides a random individual-based position updating strategy for preventing an algorithm from falling into local optimum. The implementation method of the location updating strategy based on the random individuals comprises the following steps:
Figure BDA0004067406890000084
/>
each whale individual goes through the phases of surrounding predation, spiral update and hunting, after which each individual updates its position again by a position update strategy based on random individuals, taking the optimal position before and after the update.
Embodiment one:
the 23 reference test functions which are widely used are selected and comprise 7 30-dimensional unimodal test functions, 6 30-dimensional multimodal test functions and 10 fixed-dimensional multimodal test functions for testing, and the correlation functions and the expressions thereof are shown in tables 1, 2 and 3. The unimodal test function has only one optimal solution and is typically used to test the development capabilities of the optimization algorithm. The multimodal test function is characterized by multiple optima, often not easily found, so it is used to detect the ability of the optimization algorithm to explore and jump out of local optima. The fixed dimension multimodal test function has a large number of optimal solutions, but because the dimension is lower, the optimal solution is easier to find, and can be used for testing the stability of an optimization algorithm.
In order to verify the effectiveness and advantages of the improved whale optimization algorithm (QRWOA), the optimization result of the QRWOA is compared with three other optimization algorithms, wherein the three optimization algorithms are respectively as follows: traditional Whale Optimization Algorithm (WOA), differential optimization algorithm (DE) and particle swarm optimization algorithm (PSO). The comparison results are shown in Table 4.
TABLE 1
Figure BDA0004067406890000091
TABLE 2
Figure BDA0004067406890000092
TABLE 3 Table 3
Figure BDA0004067406890000101
Experimental setup and results analysis
The population number is set to 30, the maximum iteration number is set to 300, 40 experiments are performed on each test function, and the final result is presented in the form of average value and standard deviation. The average value represents the convergence performance of the optimization algorithm, and the closer the average value is to the minimum value of the real reference function, the better the convergence performance of the algorithm is. The standard deviation reflects the deviation degree of the algorithm result relative to the average, and the smaller the standard deviation is, the lower the discrete degree of the experimental result is reflected, and the better the stability is.
TABLE 4 Table 4
Figure BDA0004067406890000102
Table 4 gives a comparison of the results of several algorithms on the optimization of the test functions. From table 4 it can be seen that QRWOA is superior to other optimization algorithms both in terms of optimizing performance as well as stability performance. The unimodal test function (F1-F7) has only one optimal solution, and the optimal solution obtained by the QRWOA is closer to the global optimal solution compared with other optimization algorithms. The multimodal test function (F8-F13) differs from the previous unimodal test function (F1-F7) in that it contains a number of locally optimal solutions, and the number of locally optimal solutions increases exponentially with the number of set variables. QRWOA compares to other algorithms, with the exception that the result of test function F13 is less than DE, the other results are all optimal. A fixed dimension multimodal test function (F14-F23) which also has a number of optimal solutions. But due to the lower dimensionality, the optimal solution is easily found. The standard deviation of the QRWOA is smaller, and the stable performance is better.
In summary, compared with the basic optimization algorithm, the improved optimization algorithm has improved solving precision and stability.
Embodiment two:
the present embodiment applies the method to channel estimation value optimization in channel estimation.
Wireless communication systems typically require detailed channel information to be obtained at the receiving end to ensure proper demodulation of the transmitted signal, and channel estimation is an important component of the wireless communication system. Channel estimation is a process of estimating model parameters of a certain channel model to be assumed from received data. The traditional channel estimation method comprises a least square method channel estimation method LS and a least mean square error channel estimation method MMSE. The LS algorithm has a simple structure, can accurately perform channel estimation, but is easily affected by signal to noise ratio; MMSE algorithms are not susceptible to signal-to-noise ratio. In this embodiment, we first combine LS algorithm and MMSE algorithm to construct a new channel estimation value
Figure BDA0004067406890000111
There are two unknown parameters x in the constructed channel estimate 1 、x 2 To describe the characteristics of the transmitted signal. To obtain a more accurate channel estimate, a modified whale optimization algorithm is used to select the appropriate parameter x 1 、x 2 The MSE (mean square error) of the estimated channel and the actual channel is minimized. The optimization model for channel estimation is as follows:
Figure BDA0004067406890000112
Figure BDA0004067406890000113
Figure BDA0004067406890000114
Figure BDA0004067406890000115
-1＜x 1 ＜1，-1＜x 2 ＜1；
wherein ,
Figure BDA0004067406890000121
representing the channel estimate obtained by improving the whale optimization algorithm, < >>
Figure BDA0004067406890000122
For channel estimation using least squares algorithm LS +.>
Figure BDA0004067406890000123
For channel estimation using minimum mean square error algorithm (MMSE), H is the real channel, +.>
Figure BDA0004067406890000124
Is a cross-correlation matrix, R HH Is the autocorrelation matrix of the real channel. I is an identity matrix, X is a transmission signal matrix, Y is a reception signal vector, +.>
Figure BDA0004067406890000125
For the power of the transmitted signal, +.>
Figure BDA0004067406890000126
Is the power of the noise signal.
Fig. 2 shows a plot of MSE convergence obtained over 30 iterations. From the convergence curve, it can be seen that the MSE can be quickly converged to the minimum value by using the method provided by the invention, so as to obtain the optimal channel estimation value.
Embodiment III:
as shown in fig. 3, this embodiment applies this method to the design of the compression spring.
The compression spring design can be modeled as an optimization problem with the goal of minimizing its mass f (x) under certain constraints, including minimum deflection, strong shear stress, oscillation frequency, and outer diameter limitations of 4 inequality constraints, the average diameter of the spring coil D (x 1 ) Diameter of spring wire d (x) 2 ) The number of effective turns of the spring N (x 3 ) 3 design variables, as shown in fig. 3.
The optimization problem described above can be solved by the following model:
Figure BDA0004067406890000127
/>
Figure BDA0004067406890000128
Figure BDA0004067406890000129
Figure BDA00040674068900001210
Figure BDA00040674068900001211
Figure BDA00040674068900001212
Variable range 0.05≤x 1 ≤2.00，
0.25≤x 2 ≤1.30，
2.00≤x 3 ≤15.0.
by adopting an improved whale optimization algorithm, the results shown in the table 5 can be obtained through 40 experiments and solving of 300 times at most, namely, the optimal spring design parameters meeting the requirements are obtained: average diameter of spring coil D (x) 1 ) Diameter of spring wire d (x) 2 ) And the effective number of turns of the spring N (x 3 )。
TABLE 5
Figure BDA0004067406890000131
It can be further seen from table 6 that the solution method (QRWOA) proposed by the present invention can effectively solve 3 design variables, and the mean value and standard deviation of the solution result are superior to the original whale optimization algorithm (Whale Optimization Algorithm, WOA), particle swarm optimization algorithm (Particle Swarm optimization, PSO) and gravity search algorithm (Gravitational Search Algorithm, GSA).
TABLE 6
Figure BDA0004067406890000132
The above description is only a preferred embodiment of the present invention, and is not intended to limit the invention in any way, and any person skilled in the art may make modifications or alterations to the disclosed technical content to the equivalent embodiments. However, any simple modification, equivalent variation and variation of the above embodiments according to the technical substance of the present invention still fall within the protection scope of the technical solution of the present invention.

Claims (6)

1. An optimization problem solving method based on an improved whale optimization algorithm is characterized by comprising the following steps:
based on the whale optimization algorithm, an improved whale optimization algorithm based on quasi-reflection learning is constructed, and the method comprises the following steps:
1) Initializing a population based on quasi-reflection learning;
2) Optimizing whale individuals in the population by adopting nonlinear convergence factors, and updating the individual positions;
3) Adopting a random individual-based position updating strategy to secondarily update the positions of the population individuals;
and solving the engineering field optimization problem by using the obtained improved whale optimization algorithm.
2. The optimization problem solving method based on the improved whale optimization algorithm according to claim 1, wherein in the step 1), the implementation method for initializing the population based on quasi-reflection learning is as follows:
Figure FDA0004067406880000011
wherein ,
Figure FDA0004067406880000012
representing the position of the i-th whale in the j-dimensional space,/I>
Figure FDA0004067406880000013
and />
Figure FDA0004067406880000014
Respectively indicate->
Figure FDA0004067406880000015
And rand (0, 1) is a random number between 0 and 1.
3. The optimization problem solving method based on the improved whale optimization algorithm according to claim 1, wherein in the step 2), the implementation method for optimizing whale individuals in the population by using nonlinear convergence factors is as follows:
the optimizing process comprises three optimizing modes, namely surrounding a prey, foaming net attack and searching predation;
surrounding the prey: suppose that in d-dimensional space, the current best whale individual X * The position of (2) is
Figure FDA0004067406880000016
Whale individual X j The position of (2) is +.>
Figure FDA0004067406880000017
The next position X of the whale individual under the influence of the optimal position j+1 Is that
Figure FDA0004067406880000018
The specific mathematical model is as follows:
Figure FDA0004067406880000019
Figure FDA00040674068800000110
C 1 ＝2r 2
A 1 ＝2a·r 1 -a
Figure FDA00040674068800000111
wherein ,
Figure FDA00040674068800000112
representing the space coordinate X j+1 The kth component, r 1 and r2 Are all random numbers between 0 and 1, max iter The maximum iteration number is t, and the current iteration number is t;
foaming net attack: two mathematical models were constructed to simulate this predation behavior, respectively:
a) Shrink wrapping; the mathematical model constructed by the predation behavior is almost identical with the mathematical model surrounding the prey, and one difference is A 1 Is a value range of (a); since the contracted envelope is to bring the whale individual at the current position close to the whale individual at the current optimal position, A 1 The value of (a) is in the range from original [ -a, a]Adjust to [ -1,1]Other formulas remain unchanged;
b) Spiral position updating; the predation behavior is that the current whale individual approaches to the current optimal whale individual in a spiral mode, and a specific mathematical model is as follows:
Figure FDA0004067406880000021
Figure FDA0004067406880000022
/>
wherein b is a logarithmic spiral shape constant, and l is a random number between-1 and 1;
the whale in the seat not only shrinks the wrapping ring, but also walks in a spiral form towards the prey, so that the position update chooses either to shrink the wrapping ring or to walk in a spiral form towards the prey with 50% probability, respectively, the specific mathematical model is as follows:
Figure FDA0004067406880000023
searching for predation: in a mathematical model of contraction surrounding this-predatory behaviour A 1 The range of the value of (C) is limited to [ -1,1]But when A 1 The value of (2) is not within [ -1,1]At this time, the current whale individual may not approach the current best whale individual, but randomly select one whale individual from the current whale population; suppose that in d-dimensional space, one whale individual X is randomized in the current whale population rand The position of (2) is
Figure FDA0004067406880000024
Whale individual X j The position of (2) is +.>
Figure FDA0004067406880000025
The mathematical model of search predation behavior is as follows:
Figure FDA0004067406880000026
Figure FDA0004067406880000027
C 1 ＝2r 2
A 1 ＝2a·r 1 -a
Figure FDA0004067406880000028
4. the optimization problem solving method based on the improved whale optimization algorithm according to claim 1, wherein in the step 3), the implementation method of the location updating strategy based on the random individuals is as follows:
Figure FDA0004067406880000031
each whale individual goes through the phases of surrounding predation, spiral update and hunting, after which each individual updates its position again by a position update strategy based on random individuals, taking the optimal position before and after the update.
5. The optimization problem solving method based on the improved whale optimization algorithm as claimed in claim 1, wherein the method is applied to channel estimation value optimization in channel estimationThe method comprises the steps of carrying out a first treatment on the surface of the Firstly, combining LS algorithm and MMSE algorithm to construct a channel estimation value
Figure FDA0004067406880000032
There are two unknown parameters x in the constructed channel estimate 1 、x 2 To characterize the transmitted signal; to obtain a more accurate channel estimate, a modified whale optimization algorithm is used to select the appropriate parameter x 1 、x 2 To minimize the mean square error MSE of the estimated and actual channels; the optimization model for channel estimation is as follows:
Figure FDA0004067406880000033
Figure FDA0004067406880000034
Figure FDA0004067406880000035
Figure FDA0004067406880000036
-1＜x 1 ＜1，-1＜x 2 ＜1
wherein ,
Figure FDA0004067406880000037
representing the channel estimate obtained by improving the whale optimization algorithm, < >>
Figure FDA0004067406880000038
For channel estimation using least squares algorithm LS +.>
Figure FDA0004067406880000039
For channel estimation using the minimum mean square error algorithm MMSE, H is the true channel,
Figure FDA00040674068800000310
is a cross-correlation matrix, R HH Is the autocorrelation matrix of the real channel; i is an identity matrix, X is a transmission signal matrix, Y is a reception signal vector, +.>
Figure FDA00040674068800000311
For the power of the transmitted signal, +.>
Figure FDA00040674068800000312
Is the power of the noise signal. />
6. The optimization problem solving method based on the improved whale optimization algorithm according to claim 1, wherein the method is applied to the design of compression springs; modeling the design of a compression spring as an optimization problem with the design objective of minimizing its mass f (x) under certain constraints, including minimum deflection, strong shear stress, oscillation frequency, and outer diameter limitations of 4 inequality constraints, the average diameter of the spring coil D (x 1 ) Diameter of spring wire d (x) 2 ) The number of effective turns of the spring N (x 3 ) 3 design variables; the correlation model is as follows:
Figure FDA0004067406880000041
Figure FDA0004067406880000042
Figure FDA0004067406880000043
Figure FDA0004067406880000044
Figure FDA0004067406880000045
Figure FDA0004067406880000046
Variable range 0.05≤x 1 ≤2.00
0.25≤x 2 ≤1.30
2.00≤x 3 ≤15.0
solving to obtain the optimal spring design parameters meeting the requirements by improving a whale optimization algorithm: average diameter of spring coil D (x) 1 ) Diameter of spring wire d (x) 2 ) And the effective number of turns of the spring N (x 3 )。
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CN116709240A (en) * 2023-06-26 2023-09-05 南京理工大学 Hierarchical sensor deployment method based on whale optimization algorithm
CN116721433A (en) * 2023-06-08 2023-09-08 吉首大学 Improved whale optimization algorithm and application method thereof in character recognition
CN117434829A (en) * 2023-12-21 2024-01-23 济南大学 Aircraft main engine wheel fan PID control method based on improved Jin Chai algorithm

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Publication number Priority date Publication date Assignee Title
CN116721433A (en) * 2023-06-08 2023-09-08 吉首大学 Improved whale optimization algorithm and application method thereof in character recognition
CN116721433B (en) * 2023-06-08 2024-01-30 吉首大学 Improved whale optimization algorithm and application method thereof in character recognition
CN116709240A (en) * 2023-06-26 2023-09-05 南京理工大学 Hierarchical sensor deployment method based on whale optimization algorithm
CN116709240B (en) * 2023-06-26 2024-03-12 南京理工大学 Hierarchical sensor deployment method based on whale optimization algorithm
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