An improved MOEA/D with dual objective-reduction method for antenna design

Abstract

Due to the high requirements for performance of antennas in the modern communication system, more and more objectives need to be considered, which makes the antenna design unendurable time-consuming and hard to converge. To obtain the Pareto solutions for antenna design effectively and efficiently, a dual reduction method is incorporated in an improved multi-objective evolutionary algorithm based on decomposition (MOEA/D). To obtain information about conflicting objectives on the Pareto front, the size of the MOEA/D neighbourhood is dynamically adjusted; to improve the speed and accuracy of finding redundant objectives, PCA and Pareto impact ratio are used as dual objective-reduction criteria. Numerical results in solving the mathematical benchmark problems, an antenna array problem and Yagi-uda optimal design demonstrate that the proposed method can reduce redundant objectives and increase the convergence speed.

Keywords

MOEA/D dual objective-reduction method inverse problem antenna design

1. Introduction

The antenna is one of the most important components of a wireless communication system, which has a critical impact on the performance of the entire system [1]. In recent years, optimization algorithms are proposed for antenna designs and electromagnetic inverse problems [2–8]. In the development of 5G and 6G communication systems, essential objectives such as multiple operation bands, high gain, low loss, and small size need to be considered synchronously and complex structures of antennas are applied to achieve the goals. Therefore, modern antenna designs are often involved high dimensions on both objectives and decision variables, which leads to strong nonlinearity and local optimal solutions. As the objective number increases, most of the multi-objectives optimization algorithms are suffering three major problems: (1) the proportion of current non-dominated individuals in the population increases dramatically in the evolutionary iteration, which tends to make the algorithm in stagnancy; (2) the number of individuals required to search the Pareto front grows exponentially; (3) difficulty in visualizing the objective space causes inconvenience to decision makers. In addition, the objectives in the actual antenna design do not always conflict with each other, it is necessary to reduce the redundant objectives. In this regard, an improved multiobjective evolutionary algorithm based on decomposition (MOEA/D) method with a dual reduction method is proposed in this paper.

2. An improved MOEA/D with a dual objective-reduction method

MOEA/D divides a multi-objective optimization into several single-objective optimization sub-problems. The weight vector 𝜆 is used to aggregate the multi-objectives into single objectives, which is preselected and kept fixed in the evolutionary iterations. The sub-problems can be described as [9]: $\begin{eqnarray}\displaystyle g^{te}(x\mid {\lambda}^{j},z^{\ast })=\max _{1\leq i\leq m}∼\{{\lambda}_{i}^{j}|f_{i}(x)-z_{i}^{\ast }|\}\quad (j=1,2,\ldots ,N) & & \displaystyle\end{eqnarray}$ (1) where, ${\lambda}^{j}∼=∼({\lambda}_{1}^{j},\ldots ,{\lambda}_{m}^{j})^{T}$ is the weight vector of the jth sub-problem; $z^{\ast }∼=∼(z_{1}^{\ast },\ldots ,z_{m}^{\ast })^{T}$ is a vector of the reference points; $z^{\ast }∼=∼\min \{f_{i}(x)\mid x\in X\}$ , for each i = 1,2, …, m; N is the number of sub-problems and m is the number of objectives in the optimization problem.

In MOEA/D, neighbourhood vector B is defined for replacement and selection in sub-problems, selected based on Euclidean distance between weight vectors. For sub-problem j, the neighbourhood vector can be recorded as $B(j)∼=∼\{j_{1},\ldots ,j_{T}\},{\lambda}^{j1},\ldots ,{\lambda}^{jT}$ are the T closest weight vector to 𝜆^j, the distance between weight vector 𝜆^j and 𝜆^k can be calculated as: $\begin{eqnarray}\displaystyle d_{jk}=\sqrt{({\lambda}_{1}^{j}-{\lambda}_{1}^{k})^{2}+\cdots +({\lambda}_{m}^{j}-{\lambda}_{m}^{k})^{2}}. & & \displaystyle\end{eqnarray}$ (2) An external population (EP) is used to store non-dominated solutions during the search. The brief updating procedure of MOEA/D is that for each sub-problem new solution is generated based on its neighbourhood vector and the best solutions on each objective function, neighbouring solutions and EP are updated by comparing the new solutions with the original ones. The evolution of sub-problems and information exchange based on neighbourhood vectors push the optimization moving towards convergence. MOEA/D optimizes all the objective functions synchronously in a single run.

2.1. Improvement of MOEA/D

As aforementioned, the neighbourhood vector B is used for selection and replacement in sub-problems, which determines the selection range of parents and the replacement range of offspring. In MOEA/D, the size of the neighbourhood vector T is fixed, which is easy to implement in the algorithm. However, if the size of B is too large, the parents are selected from a wide range, the algorithm cannot focus on the promising area; if it is too small, an excessive number of similar offspring may be generated, and the algorithm tends to fall into a local optimum. Therefore, a dynamic size for the selected neighbourhood and replacement neighbourhood is proposed to fulfil the different demands of evolution in different searching stages. The selection neighbourhood should be larger and the replacement neighbourhood should be smaller in the early stages, which benefits maintaining the diversity of the population and exploring more objective space; the situation is reversed at the end of the algorithm to increase the convergence speed. The size of the selected neighbourhood and replacement neighbourhood are updated as: $\begin{eqnarray}\displaystyle T_{s}^{i}\text{} & =\text{} & \displaystyle \max \left(\left(1-\frac{G_{i}}{G_{\max }}\right)T_{\max },T_{\min }\right)\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle T_{r}^{i}\text{} & =\text{} & \displaystyle \frac{G_{i}}{G_{\max }}T_{\max }.\end{eqnarray}$ (4) Where $T_{s}^{i}$ and $T_{r}^{i}$ are the size of selection and replacement neighbourhood in ith iteration. T_max and T_min are the upper and lower limits, respectively. G_max is the maximum iteration number, and G_i is the current iteration number in ith iteration.

2.2. Dual objective-reduction method

To remove the redundant objective precisely, a dual objective-reduction method based on PCA [10] and Pareto impact ratio [11] is proposed. The current population and objectives are processed based on PCA first. The Pareto objective is scalarized and the covariance matrix V and the correlation coefficient matrix R are calculated [10]: $\begin{eqnarray}\displaystyle V_{ij}=\frac{X_{i}X_{j}^{T}}{M-1}\quad R_{ij}=\frac{V_{ij}}{\sqrt{V_{ij}\cdot V_{jj}}}. & & \displaystyle\end{eqnarray}$ (5) Where X_i is the ith objective value vector, V_ij is the covariance between X_i and X_j, and R_ij denotes the correlation coefficient between X_i and X_j. M is the number of samples.

The matrix of eigenvectors is calculated based on matrix R obtained with (5) and sorted based on their corresponding eigenvalues. The principal components are designated with the matrix of eigenvectors. The value of ith element in the principal component reflects the contribution of ith objective to the principal component. Therefore, conflicting objectives can be obtained by selecting the elements with the larger difference value in the principal components.

To guarantee the redundant objectives are found properly, the conflicting objectives searched by PCA are analyzed using Pareto relation analysis [11]. Pareto impact ratio is used to measure the influence of one objective on the current Pareto solutions, computed as [11]: $\begin{eqnarray}\displaystyle R_{f}=1-\frac{N_{F-f}}{N_{F}}. & & \displaystyle\end{eqnarray}$ (6) Where R_f is the Pareto impact ratio of objective f, N_F represents the number of non-dominated solutions in the original objective set and N_F−f represents the number of non-dominated solutions after removing objective f.

The larger value of R_f one objective could get, the more impact on the Pareto solutions caused by this objective. When the Pareto impact ratio is lower than a predefined threshold, it is considered that all the redundant objectives are found.

2.3. Framework of the algorithm

To facilitate the implementation the flow chart of the proposed method is demonstrated in Fig. 1 and the iterative process is given as follows.

Fig. 1.

Flow chart of the proposed algorithm.

Step 1: Define and initialize global variables of the algorithm: m the number of objectives; t the iteration number; I ₀ the initial objective set; T_C threshold of PCA; R_f0 threshold of Pareto impact ratio. Generate the initial population P₀.

Step 2: The improved MOEA/D algorithm is embedded in the main algorithm as a sub-program:

Define and initialize the local variables of MOEA/D: the weight vector 𝜆; z best value so far of each objective; the boundary of neighbourhood T_max and T_min; G_max the maximum iteration number; i the iteration number; EP external population.

Calculate distances between weight vectors with (2) to obtain selection and replacement neighbourhood vectors B _s and B _r for each sub-problem. Calculate the objective function value and the aggregation function value using (1).

While i < G_max, update neighbourhood vectors B _s and B _r with T_s and T_r based on (3) and (4). For each sub-problem, select the parents from the B _s and generate new solutions calculate the objective function value and the aggregation function value of the new solutions. Update z with the best objective function value so far. Update sub-problem best value within the B_r if new solutions attain better aggregation value. Update EP by removing all the vectors dominated by new solutions and adding new solutions if no vectors in EP dominate new solutions.

Step 3: Implement PCA on EP the output from MOEA/D to have a non-redundant objective set IR_t. IR_t is further analyzed based on the Pareto impact ratio computed with (6) to obtain the final non-redundant objective set $IR_{t}^{\prime }$ .

Step 4: If $IR_{t}∼=∼IR_{t}^{\prime }$ , the objective-reduction operation is considered finished, go to Step 5 and EP is outputted as the final solution; otherwise, EP is added to the current population P_t, use $IR_{t}^{\prime }$ as the current objective set I_t, go to Step 2.

Step 5: Stop algorithm.

3. Simulation results

To verify the performance of the proposed algorithm, it is used to solve the DTLZ2 and DTLZ5 [12], the optimal design of the antenna array [13] and a microstrip Yagi-uda antenna optimization.

3.1. DTLZ test functions

DTLZ2 and DTLZ5 are used to test the performance of the proposed algorithm in optimization problems. DTLZ5 (I, M) is a test problem with redundant objectives, in which I determine the number of non-redundant objectives. The main parameters of the proposed algorithm are set as: population size N = 400, T_C = 0.95, R_f0 = 0.85, T_max = 100, and T_min = 20. Table 1 compares the proposed algorithm to the PCA-NSGA-II [10] in terms of success rate in finding the redundant objective within 20 independent runs, as well as the inverted generation distance (IGD), which is used to measure the distance between a searched Pareto solution and the true one. Table 2 demonstrates the true non-redundant objectives and iteration numbers for the dual objective-reduction method and PCA to remove the redundant objectives correctly.

Table 1
Comparison of success rate in finding redundant objectives and IGD

Test problems Success rate Test problems IGD

PCA-NSGA-II Proposed NSGA-II MOEA/D Proposed

DTLZ5 (2,10) 18/20 20/20 DTLZ5 (2,5) 0.0915 0.0914 0.0041

DTLZ5 (3,10) 16/20 19/20 DTLZ5 (2,10) 0.2742 0.0642 0.0023

DTLZ5 (5,10) 8/20 15/20 DTLZ5 (2,15) 0.1542 0.0751 0.0022

DTLZ2 (5) 20/20 20/20 DTLZ2 (5) 0.2451 0.3125 0.2413

DTLZ2 (10) 15/20 20/20 DTLZ2 (10) 2.4214 0.7152 0.8372

DTLZ2 (15) 13/20 17/20 DTLZ2 (15) 2.1426 1.2226 0.9328

Test problems	Success rate	Test problems	IGD
DTLZ5 (2,10)	18/20	20/20	DTLZ5 (2,5)	0.0915	0.0914	0.0041
DTLZ5 (3,10)	16/20	19/20	DTLZ5 (2,10)	0.2742	0.0642	0.0023
DTLZ5 (5,10)	8/20	15/20	DTLZ5 (2,15)	0.1542	0.0751	0.0022
DTLZ2 (5)	20/20	20/20	DTLZ2 (5)	0.2451	0.3125	0.2413
DTLZ2 (10)	15/20	20/20	DTLZ2 (10)	2.4214	0.7152	0.8372
DTLZ2 (15)	13/20	17/20	DTLZ2 (15)	2.1426	1.2226	0.9328

As shown in Table 1, the proposed algorithm can find the redundant objectives and has a higher success rate in finding the redundant objectives compared to PCA-NSGA-II. The advantage of the proposed method is more obvious as the number of redundant objectives increases. From the aspect of IGD values, the solutions obtained by the proposed method have the smallest IGD on solving DTLZ5 and competitive results on DTLZ2. From Table 2, it is known that the iterations needed by the proposed dual reduction method are less than PCA on finding non-redundant objectives which will save time consumption of the algorithm. In DTLZ5(3,10) and DTLZ(5,10), f₁ − f₈ and f₁ − f₆ are designed as non-conflict objectives, anyone in the non-conflicted objectives can represent the group. So it is equivalent to obtain f₇f₉f₁₀ and f₈f₉f₁₀, f₂f₇f₈f₉f₁₀ and f₆f₇f₈f₉f₁₀ in DTLZ5(3,10)and DTLZ(5,10) as final non-redundant objective set.

Table 2

Comparison between dual objective-reduction method and PCA

Test problems	PCA		Dual objective-reduction method
	Result	Iterations	Result	Iterations
DTLZ5(2,10)	f ₉ f ₁₀	3500	f ₉ f ₁₀	2500
DTLZ5(3,10)	f ₇ f ₉ f ₁₀	3000	f ₈ f ₉ f ₁₀	1000
DTLZ5(5,10)	f ₂ f ₇ f ₈ f ₉ f ₁₀	3000	f ₆ f ₇ f ₈ f ₉ f ₁₀	1000

3.2. Linear antenna array optimization

To demonstrate the proposed method in antenna array designs, a 10-element linear antenna array is optimized based on the spacing and excitation amplitude of array elements. As shown in Fig. 2, the design variables are the distance d_i between point sources and the excitation amplitude A_i, i = 1, …,5. The antenna design has six objectives: directionality, side lobe level (SLL), deep zeros at 20°, 40°, 50°, and peak side lobe ratio (PSLR), recorded as f₁, f₂, f₃, f₄, f₅, f₆, respectively.

Fig. 2.

Schematic diagram of 10-element linear antenna array.

In this case study, the parameters of proposed method are set as: population size N = 150, T_c = 0.95, R_f0 = 0.9, T_max = 40, T_min = 7. The original objective set is $I_{0}∼=∼\{f_{1},f_{2},f_{3},f_{4},f_{5},f_{6}\}$ . Through the improved MOEA/D with dual objective-reduction method, f₆ PSLR is considered as a redundant objective, and the final non-redundant objective set is $I∼=∼\{f_{1},f_{2},f_{3},f_{4},f_{5}\}$ .

Table 3

The smallest and largest values obtained by the two algorithms are compared with the uniform array

	−20log D	SLL	\|AF (20°)\|	\|AF (40°)\|	\|AF (50°)\|	PSLR	Better than uniform number
Uniform	−22.1	−15.1	−31.2	−45.9	−27.3	−12.9
The proposed	−24.1	−70.9	−130.7	−139.8	−131.2	−18.7	142
	−22.8	−32.7	−75.1	−76.2	−28.8	−12.8
PCA-NSGA-II	−23.7	−38.9	−110.3	−114.4	−104.2	−18.6	49
	−20.8	−14.8	−24.3	−13.6	−17.8	−8.3

Uniform antenna is taken as the benchmark solution; the optimization solutions searched by the proposed method and PCN-NSGA-II are compared. In Table 3, the smallest value and the largest value on each objective and the number of solutions better than benchmark solutions are given. From Table 3, it is shown that the proposed method has a strong searching ability. Table 4 gives one typical solution and its corresponding variables obtained by the proposed method and PCA-NSGA-II, and the solution obtained by the proposed method can dominate the one by PCA-NSGA-II, which proves the performance of the proposed algorithm.

Table 4

Representative optimization solution for antenna arrays

			−20 log D	SLL	\|AF (20°)\|	\|AF (40°)\|	\|AF (50°)\|	PSLR
	The proposed		−23.6	−37.3	−63.9	−123.8	−126.7	−13.8
	PCA-NSGA-II		−23.3	−24.1	−48.4	−108.8	−30.5	−13.2
	d ₁	d ₂	d ₃	d ₄	d ₅	A ₁	A ₂	A ₃	A ₄	A ₅
The proposed	0.36	1.15	1.95	2.75	3.49	0.29	0.37	0.36	0.29	0.33
PCA-NSGA-II	0.41	1.41	2.28	3.15	3.93	0.58	0.47	0.38	0.47	0.22

3.3. Microstrip Yagi-uda antenna optimization

The proposed method is used to optimise a microstrip Yagi-uda antenna. The structure of the microstrip Yagi-uda antenna is shown in Fig. 3, which is composed of an excitation element, deflector and reflection patch. The design variables are: (1) L_arm represents the length of the excitation arm, 0. 21𝜆 < 0. 5L_arm < 0. 38𝜆; (2) L_def refers to the length of deflector, 0. 41𝜆 < L_def < 0. 73𝜆; (3) D_r represents the length of microstrip line, 0.22𝜆 < D_r < 0.33𝜆; (4) D_i (i = 1, …,5) represents the distance between deflectors, 0.13𝜆 < D_i < 0.22𝜆. The central frequency is 2.45 GHz and the objectives are: return loss, wave ratio (VSWR), peak gain, front-to-back ratio, resistance and reactance in matching impedance, recorded as f₁, f₂, f₃, f₄, f₅, f₆, respectively.

Fig. 3.

Schematic diagram of microstrip Yagi-uda antenna.

In this case study, the parameters of proposed method are set as: population size N = 100, T_c = 0.95, R_f0 = 0.9, T_max = 25, T_min = 5. The original objective set is $I_{0}∼=∼\{f_{1},f_{2},f_{3},f_{4},f_{5},f_{6}\}$ . Using the proposed method, f₂, f₅ and f₆ are considered redundant objectives, and the final non-redundant objective set is $I∼=∼\{f_{1},f_{3},f_{4}\}$ . Based on the PCA-NSGA-II method, f₆ is regarded as a redundant objective, and the final non-redundant objective set is $I∼=∼\{f_{1},f_{2},f_{3},f_{4},f_{5}\}$ . The proposed method can find more redundant objectives. Table 5 shows the optimal solution obtained by the proposed method and PCA-NSGA-II. It can be seen that the proposed algorithm can find a qualified solution for the practical antenna design, and the solution provide by the proposed method is better than PCA-NSGA-II.

Table 5

Representative optimization solution for Yagi-uda antenna

		S11	VSWR	-peak Gain	Front-to-back ratio	\|50 − Re(Z_in)\|	\|Im(Z_in)\|
The proposed		−52.2	1.004	−8.8	−28.8	0.23	0.06
PCA-NSGA-II		−38.1	1.027	−8.6	−23.7	0.53	1.01
	L _arm	L _def	D _r	D ₁	D ₂	D ₃	D ₄	D ₅
The proposed	42.1	38.1	18.9	8.8	9.8	12.4	10.3	10.5
PCA-NSGA-II	41.5	36.5	18.3	8.1	8.6	13.6	10.7	13.4

4. Conclusions

To solve the redundant objective problem, this paper presents a dual objective-reduction method based on an improved MOEA/D. The DTLZ test function, the antenna array, and the Yagi-uda optimization problem show that the improved algorithm can find the non-redundant objective set and converge to the Pareto front.

Footnotes

Acknowledgements

This work was supported by the Natural Science Foundation, China under Grant No. 52077203 and the Fundamental Research Funds for the Provincial Universities of Zhejiang 2021YW06.

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