Every generation of eagles goes through a series of stages. This section explains DEOPSR in great depth. First, we outline the three essential steps of the process: (i) the initialization phase, which takes place just once during an eagle’s generation, (ii) the local phase and finally, (iii) the global phase. Then, this research work discusses the adaptive processes employed by using DEOPSR as a guideline.
3.1 Initialization Phase
Search space with lower and higher bounds of \({P}_{min}\)and \({X}_{max }\)for bound-constrained minimization problem with the objective function. In terms of dimension \(Y\), DEOPSR begins with a direction \(P,\) and it is the initial eagle in the overall population. Each \(P\) is provided by\({P}_{s}\).
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\({P}_{s}={P}_{min}+\left[{P}_{max}-{P}_{min}\right].1hs,\)
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(1)
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The population of eagles, in the beginning, is called\({E}_{0}\), and \(I = 1, 2,..., {E}_{0}\). Here, \({P}_{min}\) and \({P}_{min}\) and \({P}_{max}\) are 1 \(\times Y\) is used as a scalar multiplication symbol for vectors. This notation throughout the text will denote scalar products. The \(1hs\) number represents the Latin hypercube sample of \({E}_{0}\) rows and \(Y\)columns. Due to the random sampling method, DEOPSR ensures that the starting eagles are scattered evenly throughout each dimension. The eagle’s function values are then gathered and arranged logically. The eagle with the lowest function value is chosen as the best eagle in the original population, and it is referred as \({P}^{*}\).
3.2 Local Search Phase
The most successful eagle searches for food inside its region in the previous phase. DEOPSR use \({Q}^{*}\) to identify the finest food, and it will look for the same. A scalar radius \({Q}_{size }\) determines the lower and upper bounds of the territory’s \({Q}_{min }\) and \({Q}_{max }\) values. The boundaries of the territory are determined as a result of this process.
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\({Q}_{size }=max\left[\phi .min\left({P}_{max}-{P}_{min}\right),1\right],\)
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(2)
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\({Q}_{min}={P}^{*}-{Q}_{size}.\overrightarrow{1}, {Q}_{max}={P}^{*}+{Q}_{size}\).\(\overrightarrow{1},\)
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(3)
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where \(\overrightarrow{1}\) is a \(Y\)-dimensional vector of all ones.
To provide a vast area where the best eagle hunt for food, Eq. (2) uses the \(max\)operator, even when just a tiny search region is available. DEOPSR set the value of \(\phi\) to a threshold value (i.e., 0.06) in Eq. (2). DEOPSR sees the territory’s boundaries as a limited search space if it crosses beyond the boundaries.
Eagles in DEOPSR apply its interior-point technique, the most efficient way to find food. Consider \({P}^{*}\) as a starting point which assigns maximum function values to \({E}_{loc}\) and it treats them as the initial parameter. DEOPSR utilizes range boundaries \({Q}_{min}\)and \({Q}_{max}\) to define the range of communication. It claims that the mid-point strategy can improve exploitation ability. Once the best eagle identifies the best food, then the Global Phase immediately begins by creating a new eagle population. This new population’s best eagle will be chosen for performing local food search, and it is sorted using their function values. For each generation of eagles, an individual eagle performs the hunting for food in their local search area. Therefore, DEOPSR uses exploitation substantially to speed up the optimization process. The eagle’s territorial behavior might be depicted using this local exploitation approach as an influence. Another interesting factoid to consider is that if an \({P}^{*}\) is chosen by two successive generations, then \({Q}^{*}\) will be the starting point for the subsequent generation’s mid-point approach.
3.3 Global Search Phase
A probability vector, designated as \(M\), is used to partition the eagle population into three subpopulations after the Local Phase. An operator is assigned to each subpopulation to move or replace the eagles with the fresh ones. The freshly initialized eagles are referred to as eagle offspring, denoted by\({ P}_{new}\) after applying the relevant operators. \({E}_{0}\) rows and \(Y\) columns are the same in \({ P}_{new}\) as they are in their \(P\). It is essential to keep in mind that only the eagles with enhanced function values will be passed on to the next generation. In addition, each operator uses its parameter known as the scaling factor and designated by the letter \(G\).
Three different operators, namely (i) Movement Operator, (ii) Mutation I operator, and (iii) Mutation II operator, are present in DEOPSR.
The current generation’s eagle population is indicated using \(E\), and the three operators each have subpopulations as \({E}_{1}\), \({E}_{2}\), and \({E}_{3}\). As a result, \({E}_{1}\) + \({E}_{2}\) + \({E}_{3}\) will be equivalent to \(E\). All eagles present in DEOPSR are considered as \(Y\)-sized in each operator.
3.3.1 Movement Operator
Eq. (4) represents the Movement Operator (\(MO\)) for \(s = 1, 2,..., {E}_{1}\).
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\({\left({P}_{new}\right)}_{s}={P}_{s}+{G}_{s}.\left({P}^{*}-{P}_{s}+{P}_{b1}-{P}_{arc}\right){h}^{{-y}^{2}}\times \left({P}_{near}-{P}_{s}\right),\)
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(4)
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A random eagle from the present population \({P}_{{b}_{1}}\), is compared to \({P}^{*}\) and it is the only alternate eagle in the population.
To transform from \({P}_{s}\) to\({P}_{new}\) five eagles, namely \({P}_{s},{P}_{near, } {P}^{*},{P}_{{b}_{1}, }\)and \({P}_{arc }\) are engaged. This neighbourhood contains the global optimal point and \({P}_{near }\) is the nearest eagle to\({ P}_{s}\), whereas \({P}_{near }\)is the best eagle. \({P}_{s}\) becomes \({P}_{new}\) by moving to a better location by applying combined position vectors of the five eagles.
Randomly another eagle is selected from the existing population, and the archive \({P}_{arc}\) it with \(P\) and \({P}_{{b}_{1}, }\). \({P}_{near }\)indicates the Euclidean distance to \({ P}_{s}\) which is the eagle from the present population with the least \(y\) value. A differential Evolution strategy is utilized at the beginning of \(MO\) to avoid becoming stuck in a local minimum due to the operator’s high search capabilities. Those eagles that didn’t make further generations are kept in the external archive.
The \(MO\) has the feature of considering the proximity of nearby eagles. As the movement of an eagle depends on the position of the eagle closest to it, then incorporation is done to represent the hunting strategy of the eagle in a combined manner. As a side benefit, the phrase “subpopulation” might further split the population into smaller groups, each focusing on local solutions. This characteristic is particularly beneficial when solving multimodal problems because one of these local answers may be the optimal global solution.
3.3.2 Mutation I Operator
Eq. (5) represents the Mutation \(I\) operator given for\(s = 1, 2,..., {E}_{2}\)
\({\left({P}_{new}\right)}_{s}={G}_{s}.\left({P}_{{b}_{1}}+{P}^{*}-{P}_{{b}_{2}}\right)+E.Z\left(Y\right),\)
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(5)
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where \({P}_{{b}_{1}}\)and \({P}_{{b}_{2}}\) represents the two separate eagles chosen randomly from the existing population and which must be distinct from \({P}^{*}\) in different ways.
During this time, \(E\) is an unpredictably large \(1\times Y\) random vector (0, 1). Definition of the Lévy flight function, \(Z\), is as follows:
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\(Z\left(Y\right)=\left(\frac{0.01\times o\times \in }{{\left|r\right|}^{\frac{1}{\gamma }}}, \in \right)={\left(\frac{{\Gamma }\left(1+\gamma \right)sin\left(\frac{\pi \gamma }{2}\right)}{\gamma {\Gamma }\left(\frac{1+\gamma }{2}\right){2}^{\frac{\gamma -1}{2}}}\right)}^{\frac{1}{\gamma }}\)
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(6)
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The values \(o\) and \(r\) are chosen from the standard distributions. The Gamma function is used as the \(\gamma\) parameter, which is set to 3.4 and \({\Gamma }\left(\text{p}\right)\) by default.
Many animals and insects fly and move at a different speed, whereas DEOPSR inherits eagle characteristics for considering the same the movement of nodes. Lévy flight term has been added to simulate the eagle’s flight patterns. A Lévy flight is a random walk with different step sizes taken from a Lévy distribution.
3.3.3 Mutation II Operator
Eq. (7) represents the mutation II operator given for \(s=\text{1,2},\dots .,{E}_{3}\).
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\({\left({P}_{new}\right)}_{s}={G}_{s}.\left(\widehat{P}+ {P}^{*}-{P}_{mean}\right),\)
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(7)
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where \({P}_{mean }\)represents the average of all eagles in the current population, and \(\widehat{P}\)is a randomly generated eagle inside the constrained search area.
Eq. (7) enhances the algorithm’s exploring capabilities while simulating the perching behavior of the eagle. Eagles fly in response to the positions of other birds in their search area with the inclusion of \({P}_{mean}\).
3.5 Parameter Adaptation Schemes
Adaption strategies are employed in DEOPSR to regulate specific parameters and increase performance. The eagle population size \(E\), the probability vector \(M\), and the scaling factor \(G\) are the three variables that make up these parameters. The selection of \(E\), \(M\), and \(G\) are calculated.
3.5.1 Population Size Depreciation using Linear Algebra
A linear algebra-based population depreciation (i.e., \(E\)) is carried out during the end of each generation to ensure the trust level and mathematically shown in Eq. (8).
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\(E=⟦{E}_{0}+\left({E}_{min}-{E}_{0}\right).\frac{T}{{T}_{max}}⟧,\)
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(8)
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where \({E}_{0}\) indicates the initial population of the eagle, \(T\) represents the count of current evaluations of function, and \({T}_{max}\)represents the highest count of evaluations of function. \({E}_{min}\) is the minimum population size. \(MO\) in DEOPSR requires at least five eagles in the population, and the value of \({E}_{min}\) is set as 5. The eagles with the most significant function values are called to fulfil the required population size, and it assists in enhancing the exploitation of search space.
3.5.2 Enhancement of Population Size
The probability vector \(M\) offers three different values that aid in the subpopulation assignment of eagles. The values are all set to \(0.3\) at first. A random number \(w\) between \(0\) and \(1\) is assigned to all the eagles in the current population, and if \(w\le 0.3\), then \(MO\) will be applied to evolve it. Otherwise, if \(0.3<w\le 0.6\), then Mutation I Operator will be applied to evolve it. Otherwise, the Mutation II Operator will be used. After that, the odd ones are adjusted based on the rate of operator improvement. If \({E}_{s }\)is the size of the subpopulation corresponding to operator \(s\) where \(s\) = 1, 2, 3, then \({B}_{s }\)represents the rate of improvement.
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\({B}_{s}=\frac{{\sum }_{j=1}^{{E}_{s}}max\left(0,{g}_{old,i}-{g}_{new,i}\right)}{{\sum }_{i=1}^{{E}_{s}}{g}_{old,i}}\)
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(9)
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The function values of the current eagle and its corresponding eagle offspring are \({g}_{old,i}\) and \({g}_{new,i}\) respectively. The probability value \({M}_{s}\)for the operator, \(s\) is then adjusted using Eq. (10).
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\({M}_{s}=max\left[0.1,\text{min}\left(0.8,\frac{{B}_{s}}{{B}_{1}+{B}_{2}+{B}_{3}}\right)\right]\)
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(10)
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Eq. (10) emphasizes the best-performing operator in every generation, giving it more ways for optimization. Meanwhile, underperforming operators are given a chance to improve in future generations.
3.5.3 Adaptive Control
During the Global Phase, every eagle in the population is assigned a scaling factor (i.e., \({G}_{s}\)). A Cauchy distribution with a mean \(\vartheta {G}_{s}\) and a variance of \(0.23\) is used to construct the scaling factor. If \({G}_{s}\ge 1\) is less than one, then it is rounded to value one, and if \({G}_{s}\le 0\) is more than zero, and then regeneration is performed. The mean values \({\vartheta G}_{s}\) are taken from a specific memory with a predetermined memory size \(L\). The memory’s values are set as 0.2 while beginning the process. It’s worth noting the constants employed in adaptive control, Specifically, the variance of \(0.1\) and the setting’s initial value of \(0.23\). As a result, DEOPSR treats this value to maintain the process’s consistency. Then, a memory element is modified whenever a generation produces at least one eagle offspring with a higher function value. The scaling factors for the improved eagle offspring are recorded in a vector \(J\). The weighted Lehmer mean is utilized for synchronizing the data between different iterations.
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\({mean}_{NZ}\left(J\right)=\frac{{\sum }_{a=1}^{\left|J\right|}{N}_{a}{G}_{a}^{2}}{{\sum }_{a=1}^{\left|J\right|}{N}_{a}{G}_{a}}, {N}_{a}=\frac{\varDelta {g}_{a}}{{\sum }_{z=1}^{\left|J\right|}\varDelta {g}_{z}}\)
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(11)
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where \({G}_{a}\) denotes the \(k\)th scaling factor present in \(J\), and \({\varDelta g}_{a}\) indicates the \(a\)th eagle offspring’s change present in the function value.
The pseudocode present in Algorithm 1 summarizes the steps involved in DEOPSR.
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Algorithm: Pseudocode of DEOPSR
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Input:\(g,{P}_{min},{P}_{max},Y\)
Output:\({p}^{*},{g}^{*}\)
1. Define\({T}_{max},{E}_{0},{E}_{loc}\)
2. Set \(A\) ← 0, \(T\) ← 0
3. foreach \(s = 1, 2, 3,\) and\({M}_{s}\leftarrow 0.3\)
4. Apply Eq. (1) to create eagles at the beginning of population P
5. Update the value of \(T\) via sorting \(P\) based on the function value
6. Apply Eq. (2) and Eq. (3) to search \({Q}^{*}\) using a midpoint approach having maximum evaluations value
7. While \(T\le {T}_{max}\)do
8. Set\(A=A+1\)
9. Apply Eq. (8) to minimize the size of the population linearly to obtain \(E\) value
10. \(M\) is utilized in dividing the population into subpopulations
11. Apply Eq. (4) to Eq. (7) to generate a new population of eagles
12. Update \(T\) and sort \({P}_{new }\)to achieve the new function value
13. Update\({P}^{*}\)
14. Apply Eq. (9) and Eq. (10) to update \(M\) depending on the individual operator’s improvement rate
15. return\({p}^{*}={Q}^{*}\)
16. return\({g}^{*}=g\left({Q}^{*}\right)\)
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