Improved Hybrid Real-Binary PSO Using Modified Learning and Restarting Methods for Planar Microwave Absorber Designs

This paper proposes an improved hybrid real-binary particle swarm optimization (IHPSO) using modified learning and restarting methods to address electromagnetic optimization problems involving both continuous and discrete decision variables. The modified learning approach aims to improve the exploration and exploitation capabilities of IHPSO. A restarting mechanism is employed to randomly reposition the swarm of particles to prevent them from being stuck in the local optimum, while still directing the search process with the global best. To demonstrate the performance of the IHPSO, 23 benchmark functions are tested, and the results are compared with those of other traditional algorithms. In both unimodal and multimodal functions, IHPSO outperforms the original hybrid real-binary PSO. Finally, the proposed algorithm is used to design planar microwave absorbers, a classic hybrid real-binary electromagnetic optimization problem.


I. INTRODUCTION
Particle swarm optimization (PSO) algorithm, which has been widely used in electromagnetic optimization, can be divided into two categories based on whether the parameters of problems handled are continuous or discrete [1], [2], [3], [4], [5], [6], [7]. However, when dealing with problems that have both real and binary parameters, only binary PSO (BPSO) can be used via a binary-to-real mapping [6], [7]. Furthermore, this mapping results in the inevitable increase in the number of optimization parameters and the computational cost of binary optimization.
The associate editor coordinating the review of this manuscript and approving it for publication was Sotirios Goudos . The partial radiation structure of the antenna usually consists of pixelated square patches of metal or air that are used to change the surface current distribution of the original antenna in order to improve its radiation performance. A pixelated checkerboard metasurface backed by a metal plate with a gap is proposed in [12] where two pixelated cells are optimized using the BPSO algorithm to produce a broader RCS reduction frequency band. Actually, in the proposed design, the gap between the metasurface and ground is fixed, which can be regarded as a continuous parameter to further increase the RCS reduction bandwidth. Another common hybrid optimization problem is PMA design, which involves choosing the material and thickness of each layer to reduce the reflection coefficient of PMA. Several hybrid meta-heuristic algorithms are proposed to optimize this type of mixed-variable problem (MVP). An ant colony optimization algorithm for MVPs (ACO MV ) is proposed that integrates three solution generation mechanisms to deal with different variables [14]. The effectiveness and robustness of ACO MV are demonstrated by comparison results with other classic evolutionary VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ algorithms. A hybrid differential evolution (DE) algorithm, namely DE MV , is proposed based on hybridizing the original DE and the set-based DE for respectively evolving continuous and discrete variables [15]. A hybrid-coded human learning optimization (HLO) algorithm, named HcHLO, is proposed to tackle mix-coded problems based on the framework that real-coded parameters are optimized by the continuous linear learning operators of continuous HLO while the rest variables of problems are handled by the binary learning operators of HLO [16]. According to the experimental results, the HcHLO algorithm achieves the best-known overall performance.
A new variation of differential evolution, namely MCDE MV , is proposed to solve MVPs, which adopts a multi-strategy coevolutionary approach for adapting to the all-inclusive situations in MVPs [17]. The experiment results show MCDE MV is more competitive and efficient than other compared similar algorithms. A hybrid real-binary particle swarm optimization (HPSO) is proposed, which combines the real and binary variables into a vector denoted as a candidate, whose velocity and position are updated using the formulas defined by the traditional real-number PSO (RPSO) and binary PSO (BPSO) [9]. In addition, a hybrid algorithm (HDE) for mixed-variable antenna designs is suggested in [10]. It is based on the traditional differential evolution and boolean differential evolution for real and binary variables, respectively. However, HDE has a limited ability to handle high-dimension problems.
As is well known, the classic RPSO and BPSO are likely to suffer entrapment in the local optimum when addressing multimodal optimization problems. As a result, a number of improved methods for improving the convergence performance of both RPSO and BPSO have been developed. An orthogonal crossover operation (OCO) based on the Taguchi method (TM) is carried out for RPSO [18], [19], [22] and BPSO [7] to generate potential offspring. TM employs an orthogonal array and signal-to-noise ratio to design experiments and select the parameter values of the candidate. OCO can improve the exploitation ability and has been widely used in a variety of meta-heuristic algorithms [21], [22], [23], [24]. In general, experiments designed by the orthogonal array have full exploitation in the local hypercube search space formed by the levels of each factor, and the number of experiments is proportional to the optimization parameter. The improvement of learning schemes is another way to improve convergence speed and search capability. Two reproduction methods for continuous variables and discrete variables are proposed: the particles learn from randomly selected excellent personal best positions for the continuous part, and both historical search information and the current swarm's search information are considered for the discrete part [20]. The modified velocity updating method aims to preserve population diversity while improving local search capability. However, the social component offered by the global best is lacking in velocity updating, and the particles do not take full exploitation of the current global best.
In this paper, an improved hybrid real-binary PSO (IHPSO) is proposed to enhance the original HPSO algorithm's convergence performance and capability for the global optimum. Two novel methods are used, including learning and restarting methods. The population of the swarm is divided into two groups for velocity updating of the real part: one group's velocities are updated using the original rule, which learns from both the current global best particle and their historical best, while the other group learns from a randomly chosen personal best particle that is better than the current particle's personal best. To avoid the situation in which all particles are trapped into a local optimum and lack of ability to overstep the local optimum, the swarm position and personal best are initialized, and the global best is preserved to direct the swarm searching. The performance of the proposed algorithm is verified using 23 benchmark functions and the PMA designs, and the results are contrasted with those of HPSO and other representative algorithms. It is demonstrated that the IHPSO outperforms the HPSO since it achieves lower average values with smaller standard deviations.
In the following sections, this paper introduces a detailed explanation of the IHPSO algorithm, the benchmark functions optimized using the proposed algorithm, and its application in PMA designs.

II. THE IHPSO ALGORITHM
For a hybrid real-binary minimization problem f (⃗ x), where ⃗ x = (R 1 , R 2 , · · · , R m , B 1 , B 2 , · · · , B n ) T is a M + N dimension vector, and R m (m = 1, 2, . . . , M ) and B n (n = 1, 2, . . . , N ) represent real and binary number, respectively. The lower and upper bounds of parameter R m are denoted as lp m and up m . The traditional HPSO just simultaneously employs the velocity updating methods of the classic RPSO and BPSO to generate potential offspring. Vector ⃗ x represents a particle in the HPSO and its location is updated using (1) and (2) for the real part, and (3), (4), and (5) for the binary part, as shown below.
where v k+1 i,m and v k+1 i,n are the velocities of mth real variable and nth discrete binary variable of particle i at k + 1th iteration. p k i,m and g k n represent the personal best position found thus far by the ith particle and the global best position searched so far by the whole swarm. Two accelerating coefficients, c 1 and c 2 , are used to modify the knowledge learned through both personal and social experience. r 1 and r 2 are two random numbers uniformly distributed in the range of [0,1]. ω k+1 R is the inertia weight to balance the exploitation and exploration. For a binary variable x k i,n of particle ⃗ x i k , the velocity of x k i,n is updated by (3). Due to the discrete binary search space, a transfer function (TF) is needed to convert the velocity of x k i,n to a probability value. Two types of TFs were compared for BPSO in [25], and the comparative study revealed that the V-shaped family of TFs improved the performance of original BPSO. To verify the selection of TF in IHPSO, the optimization results of 23 benchmark functions by IHPSO with different transfer functions show that the S-shaped family of TFs significantly outperforms the V-shaped family of TFs. Moreover, there is not much difference between the average best values obtained by IHPSO with various S-shaped TFs. The number of functions with the lowest average best values found by IHPSO with four different S-shaped TFs is almost equal. Thus, the S-shaped TF (4), i.e., the S2 in [25], is adopted in the proposed algorithm. However, this TF is not the best choice for all test functions. The position of x k i,n could be ultimately updated by (5).

A. MODIFIED VELOCITY UPDATING SCHEMES
Although the HPSO algorithm has been applied to hybrid real-binary optimization problems, it has the weakness of being easily trapped in the local optimum and lacking the ability to escape from it. From the velocity updating methods (1) and (3), particles learn from their own personal best locations as well as the global best found so far. If the global best particle cannot be improved, learning from it may cause the particle to become stuck in the local optimum and reduce the diversity of the swarm. Inspired by the idea to keep the diversity of the swarm proposed in [20], the IHPSO swarm is split into two subswarms with distinct velocity updating methods for the real variable to preserve the diversity of the swarm and convergence to the global best. To preserve the convergence efficiency, the velocities of 70% of the particles in the swarm are updated by (1) in this paper, while the remaining particles employ a novel approach, as described in (6), to update their velocities.
where p k r a ,m represents the mth real variable of a personal best position p k r a that was randomly chosen from a group of personal best positions whose performance is better than ⃗ x k i . The definitions of ω k+1 R , c 1 , and r 1 are identical to (1). According to (6), the ability of R m to learn from various exceptional personal bests may enrich its potential values and increase the swarm diversity during the evolution process. As a result, the overall exploration capability of IHPSO has generally improved. Due to the fact that some particles learn from the social component, these two velocity updating strategies employed for the real part of IHPSO not only ensure to improve its exploration abilities, but also search around the global best to preserve the convergence efficiency. The discrete binary variable x k+1 i,n only has '0' and '1' values; as a result, the BPSO algorithm is susceptible to being trapped in a local minimum. Therefore, it is crucial to preserve the diversity of binary variables in the swarm. The velocity update approach (7) is applied to the binary portion of all particles, just as it was for the real variable of the proposed algorithm to maintain the swarm diversity.
where ω B is equal to constant 1, p k r b is also a randomly selected personal best as long as its performance is better than x k i , that is, r b can be the same with r a in (6). The transfer function (4) and (5) are adopted to convert v k+1 i,n to binary bit here.

B. A MODIFIED DYNAMIC INERTIA WEIGHT
Inertia weight is an important parameter to control the exploration and exploitation of the traditional RPSO and BPSO algorithms. A modified dynamic inertia weight is suggested to further balance the exploration and exploitation of the real part of IHPSO. The dynamic inertia weight ω k+1 R at the k +1th generation, which is calculated by (8) and (9), is normally distributed with various means and variances associated with the preset values of ω Rmax , ω Rmin and the calculated value of c k+1 ω using (9). For real part 5: if index = 0 then 6: Obtain ω k+1 R by (8) (6) and (2). 14: end if 15: For binary part 16: Update ⃗ v k+1 i and ⃗ x k+1 i by (4), (5) and (7). 17: Count the number N k+1 n=k−N r N n p = 0 and binary part of ⃗ p k+1 are identical to one another then 19: index=1. 20: for i = 1 to N do 21: ⃗ x k+1 i is randomly sampled, 22: ⃗ p k+1 i is set to zero vector. 23: end for 24: end if 25: Check the termination criterion. 26: end for where r 3 is a number from a standard normal distribution; s = c f ×N k+1 c is a coefficient to auxiliary alter the decrement rate. N k+1 c denotes the number of current particles that outperform their parent particles at the current iteration; c 1 ω is initialized to be equal to ω Rmax . When the value of c k+1 ω is greater than 0.45, ω k+1 R is a number from the normal distribution with the mean of ω Rmin and standard deviation of c k+1 ω − ω Rmin . c f is set to 20 in this paper in order to allow for a gradual decrease in the value of c k+1 ω . ω k+1 R is a number from the normal distribution with the mean of zero and standard deviation of ω Rmax − ω Rmin when the value of c k+1 ω is less than 0.45.

C. RESTARTING MECHANISM
When the personal best positions of all particles have remained unchanged for N r sustaining iterations and the binary string of each personal best is identical to one another, the position x k i and personal best p k i of each particle will be, respectively, randomly selected within boundaries and set to zero vector to further enhance the ability to jump out of the local optimum. The velocity v k+1 i of each particle is updated using 1 where the global best g k is preserved and the inertial weight ω k+1 R is obtained by (10).
Considering the limitation of the maximum iteration, this restarting procedure is carried out only once in the following numerical simulations.

III. EXPERIMENTAL RESULTS AND PERFORMANCE COMPARISON
The performance of the proposed algorithm is verified through the optimization of 23 benchmark functions and five-layered PMA designs. To demonstrate the efficiency of IHPSO, the optimization results are also compared with those of other well-known algorithms.

A. BENCHMARK FUNCTIONS
To demonstrate the searching ability and convergence of the proposed algorithm, 23 classic benchmark functions are tested under the condition that partial variables are represented by a binary string. For the binary part, the real variable of each function is mapped to a binary string with a quantization error less than 2 × 10 −4 . The detailed expression, including the upper bound and lower bound of each function, can be found in [26]. 100 independent runs are conducted, and the results obtained by the IHPSO are compared with those of five representative algorithms, including HPSO [9], HDE [10], binary genetic algorithm (BGA), hybrid grey wolf optimization (HGWO) [27], and binary equilibrium optimization (BEO) [28], the last three of which are effective at solving binary problems.
The statistics of 100 independent runs of each function obtained by the different algorithms are listed in Table 1. The minimal values of the maximum, minimum, mean, and standard deviation of each function obtained by six algorithms are shown in bold. R m and B n denote the number of real and binary variables, respectively. N b denotes the number of bits required to represent a real variable with a quantization error (QE). For these six algorithms, the population size and the maximum number of iterations are respectively set to 20 and 500. For the high dimension (M + N > 80) test functions, IHPSO yields the best results with the lowest mean and standard deviation values, with the exception of F8, F9, and F12. For functions with low dimensions where the maximum number of real variables of F14 to F23 is less than 3, HDE outperforms HPSO and IHPSO. F14 -F23 are fixed-dimension multimodal benchmark functions, and the number of variables and the boundaries of real variables are significantly lower than those of the other functions, implying that these fixed-dimension functions have more compact search spaces. It is different from HPSO or IHPSO in that the search process is guided by better particles, and the base vector in the mutation operator of HDE is randomly chosen from the population. As a result, HDE has a greater chance of escaping from the local minimum and producing  better results in the low-dimension search space. Statistics show that, with the exception of F12, the mean values and standard deviations of IHPSO for these 22 test functions are generally superior to those of HPSO. The average convergence curves of 12 functions with 100 independent runs using IHPSO, HPSO, HDE, BGA, HGWO, and BEO algorithms are depicted in Fig. 1. It can be observed that the global results obtained by HPSO are better than those obtained by IHPSO in the initial several iterations. It is because of velocity updating methods of HPSO that all of particles are VOLUME 11, 2023 learning from the global best g k+1 and are more likely to take full exploitation around the current g k+1 . However, IHPSO obtains the minimal average global values for both unimodal and multimodal functions, demonstrating the effectiveness of the proposed algorithm.

B. PLANAR MICROWAVE BROADBAND ABSORBER DESIGNS
PMA design is a hybrid optimization problem that aims to select the material with the appropriate thickness of each layer to minimize the reflection coefficient of an incident wave within the desired frequency band in the electromagnetic field. In this section, five-layered PMA designs are investigated, which are also tested in [8], [9], and [13], under the incidence wave with an angle of incidence θ. The geometry of five layered PMA with PEC-backed is depicted in Fig. 2. The materials database used in this paper is the same as that in [8]; the maximum thickness of each layer is 2 mm; and the fitness function is defined as follows: where R( f , θ) represents the maximum reflection coefficient in decibels for the plane incident wave with angle θ within the desired frequency band, and the detailed procedures used to calculate R( f , θ) can be found in [8] and [13]; γ is a penalty coefficient with a large value, which is set to 10 5 in this section; t i denotes the material thickness of each layer, and the maximum thickness of the PMA is represented by T d . Twelve cases that the PMA designs with variety T d and incidence angle θ within the high-frequency band (2)(3)(4)(5)(6)(7)(8) and the low-frequency band (0.2 -2 GHz) are tested by HPSO and IHPSO algorithms, and the statistics of 20 independent runs are listed in the Table 2. θ = 0 denotes the normal incidence, ''Null'' in T d column indicates γ = 0 that the PMA design has no thickness constraint. Compared with the maximum values obtained by HPSO, the results obtained by IHPSO are slightly better. Furthermore, the minimum values obtained by IHPSO are better than those of HPSO. It has been demonstrated that IHPSO outperforms HPSO in terms of exploitation capability. The detailed optimum results of  PMA designs with T d = 5 for normal incidence within the high-frequency band and T d = 3 for oblique incidence of TE polarization (θ = 40 • ) within the low-frequency band are listed in Table 3.
The lower reflection coefficients are obtained by IHPSO with lower thickness for these two cases,as shown in Table 3. For normal incidence case, when T d is set to 5 mm, the maximum reflection coefficient optimized by IHPSO is not only -1.24 dB less than that of HPSO, but the total thickness of PMA is more thinner than that of HPSO. For oblique incidence (θ = 40 • ) of TE polarization, T d is set to 3 mm. The optimal maximum reflection coefficients are -21.85 dB and -22.32 dB for HPSO and IHPSO, respectively. Only three different types of materials are employed, yet the overall thicknesses are nearly identical. Figures 3 and 4 depict the reflection coefficients and the convergence curves of cases 2 and 12, respectively. According to the convergence values, IHPSO achieves superior results after 2500th fitness function evaluations (FFEs). Due to the social component of velocity updating in (1), which includes a search guide by the global best, the global bests of HPSO at the first several iterations are better than those of IHPSO.
To evaluate the performance of the PMA in the wideband (0.2 -8 GHz), the R( f , θ) of the PMA designs with varying    T d and θ are obtained under the conditions that the maximum number of FFEs or the R( f , θ) meets the termination criterion. 20 independent runs are conducted by HPSO and IHPSO, respectively, and the statistics and total optimization time are presented in Table 4. R d ( f , θ) denotes the desired maximum reflection coefficient within the required band. The success rate (SR) is defined as the proportion of runs that achieve the R d ( f , θ) over 20 independent runs. The SRs of IHPSO are nearly 0.2 larger than those of HPSO, implying that the desired R( f , θ) is more likely obtained using IHPSO. The maximum results obtained by IHPSO are lower than those obtained by HPSO; meanwhile, IHPSO is much easier to meet the termination criterion.
It is a common problem in microwave absorber design to get better performance under wideband and wide-angle incidence conditions. A five-layered PMA within 0.2 -8 GHz and under wide-angle incidence is optimized. The fitness function can be expressed as follows.
where 0.2 GHz ≤ f i ≤ 8 GHz, 0 • ≤ θ j ≤ 40 • . F denotes the maximum reflection coefficient when the frequency ranges from 0.2 GHz to 8 GHz and the incidence angle varies from 0 • to 40 • . Table 5 displays the statistics and optimal results from 20 independent runs. The convergence curves obtained by HPSO and IHPSO are shown in Fig. 6(a), and the reflection coefficients versus the incidence angle of the optimal PMA designs are given in Fig. 6(b). All of the statistics obtained by IHPSO are superior to those obtained by HPSO, particularly the maximum value. These cases provide strong evidence that IHPSO has superior convergence and optimization capabilities for PMA designs when compared to HPSO.

IV. CONCLUSION
An improved HPSO that utilizes novel velocity updating and restarting methods to enhance convergence efficiency and the ability to escape from local optimums is proposed in this paper. Different learning methods can increase the swarm diversity while simultaneously maintaining the convergence speed. When all particles converge to a local minimum, a restarting mechanism is adopted to reset each particle's position and personal best so that it can restart the search process. The global best is held to speed up this second optimization phase. The effectiveness of IHPSO is investigated using 23 benchmark functions and PMA designs. The simulation results demonstrate that IHPSO produces superior results with fewer standard deviations than the original HPSO for unimodal and multimodal benchmark functions. In the first few iterations, however, HPSO outperforms IHPSO because it has a better chance of learning from the global best of the current iteration, whereas the velocity updating method of IHPSO has a better ability to escape from the local optimum and maintain population diversity. Meanwhile, an additional parameter c f , whose value is associated with population size and maximum iteration, must be set at the start. In general, IHPSO has the potential to outperform HPSO and may be a good candidate for hybrid real-binary optimization problems.
XINGNING JIA received the B.S. degree in information engineering from Xi'an JiaoTong University, Xi'an, China, in 2012, and the M.S. and Ph.D. degrees from the Communication University of China, Beijing, China, in 2017 and 2020, respectively.
In September 2020, he joined the School of Physics and Electronic-Electrical Engineering, Ningxia University, where he is currently a Lecturer. He has authored more than ten articles in peer-reviewed international journals and conference proceedings. His current research interests include MIMO antenna design and surrogate-assisted EM optimization. In November 2017, she joined the School of Physics and Electronic-Electrical Engineering, Ningxia University, where she is currently a Lecturer. She has authored five articles in peer-reviewed international conference proceedings. Her current research interests include 6G channel sounding and channel modeling. He has been with Ningxia University, since 1997, where he is currently a Professor with the Department of Information Engineering. His research interests include antenna design, channel modeling, and wireless communication. VOLUME 11, 2023