Incorporating aggregation trees in two-archive algorithm 2 for solving high dimensional multi-objective electromagnetic optimization

Abstract

The optimization designs of modern electromagnetic devices are often modeled as high dimensional multi-objective problems due to their high-quality demands in multiple functions and systemization. To identify and remove the redundant objectives, an interactive aggregation trees method embedded into the two-archive algorithm 2 is proposed to solve the high dimensional multi-objective optimizations. To balance the speed and the accuracy in finding the redundant objectives, conflict degree, conflict list and conflict flag are developed in an interactive aggregation trees frame. The two-archive algorithm 2 is modified to adjust the updated model obtained from the objective-reduction frame. Mathematical test functions and TEAM Workshop Problem 22 are used to test the performance of the proposed method. Numerical results demonstrate that the proposed method can remove the redundant objectives effectively and provide qualified solutions for the electromagnetic optimization designs.

Keywords

Aggregation trees method high dimensional optimization inverse problem two-archive algorithm 2

1. Introduction

As the multiple functions and high requirements for the modern electromagnetic systems, designers tend to make all critical concerns as optimization objectives during the electromagnetic designs. However, when the number of objective functions in an optimization problem is reaching more than four or five, traditional multi-objective optimization algorithms are hard to search the Pareto front. Because non-dominated solutions are exponentially increasing with the number of objectives which makes the algorithm can not converge. Two-archive algorithm 2 (Two-Arch 2) is proposed to cope with the high dimensional optimization problems, in which diversity archive (DA) and convergence archive (CA) are developed for balancing the diversity and convergence of population [1]. Due to the flexible structure, algorithms based on two archives are applied in engineer designs [2–4]. However, electromagnetic optimization designs are time consuming because of iterations on electromagnetic calculation, in which finite element analysis is often involved. It is an essential issues for electromagnetic designs to find qualified solutions within the desirable number of electromagnetic calculation. Furthermore, in engineering problems objectives may not always conflict to each other, so it is imperative to analyze correlation of objectives to remove the non-conflict or weak conflict objective functions and decrease the complexity of optimization algorithm. In this regard, an objective-reduction aggregation trees method embedded into Two-Arch 2 is proposed to solve the high dimensional electromagnetic optimization problems.

The main contribution of this work is to propose a multi-objective optimization algorithm with the capability of auto-identifying and processing redundant objectives in solving high dimensional electromagnetic optimizations. To achieve this goal, aggregation trees method is extended to an interactive aggregation trees frame based on conflict degree, conflict list and conflict flag to promote the accuracy of redundant objective identification; Two-Arch 2 is modified in structure and procedure to adapt the possible low or high dimensional evolutionary situations after combining the redundant objectives.

2. Two-arch 2 incorporating with interactive aggregation trees method

2.1. Interactive aggregation trees frame

Aggregation trees method is an effective way to calculate the conflict degree between objectives and provide the information in the form of a tree [5,6]. In Aggregation trees method, three types of conflicts are mentioned: direct, max-min and non-parametric. Direct conflict is the difference in absolute terms for objectives; max-min conflict measures the conflict between objectives using the maximum and minimum value in the preference area; non-parametric conflict computes the conflict regardless the comparability or unit between the objectives [5]. The direct conflict is preferred when the objectives are using the same units; max-min conflict is used when the range of value are known; non-parametric conflict is adopted when the objectives are non-comparable in units and values [5]. In the original aggregation trees method, the conflict degree of objectives is highly depended on the number and distribution of sampling points. If the number of sampling points is too small or the distribution of sampling points is crowded in one area, the conflict degree cannot represent the conflict between objectives. To estimate the conflict degree of objectives more precisely, a lot of sampling points are needed which increases the computational burden of the optimization. In order to grantee the accuracy of conflict evaluation and save the computational resources, an interactive aggregated trees frame with conflict degree, conflict list and conflict flag is developed.

Conflict degree based on non-parametric conflict is used to demonstrate the conflict between objectives, which is calculated with the evolutionary process of the optimization algorithm. Conflict degree can be calculated as [7]: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathit{Conflict}_{ab}^{i}=\frac{C_{ab}^{i}}{C_{\text{max}}^{i}}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle C_{ab}=\mathop{\sum }_{k}|X_{ka}^{\prime }-X_{kb}^{\prime }|\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle X_{kj}^{\prime }=R_{kj}\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle C_{\text{max}}=\mathop{\sum }_{k=1}^{n}|2k-n-1|\end{eqnarray}$ (4) Where, $\mathit{Conflict}_{ab}^{i}$ is conflict degree between the objective a and b in the i_th iteration; C_ab is non-parametric rank conflict between objective a and b; R_kj is the rank of individual X_kj within all X_{. j}. C_max is the maximum conflict, n is the number of sampling points.

Conflict list is defined to store information of objectives with minimum conflict degree in the latest iterations, which indicates the variation tendency of the objectives with minimum conflict degree. The structure and updating direction of conflict list are shown in Fig. 1. In Fig. 1, $C_{\min }^{i}$ is the minimum conflict in the i_th iteration and $Obj(1)_{\min }^{i}$ , $Obj(2)_{\min }^{i}$ are the corresponding objectives to $C_{\min }^{i}$ ; the arrows represents the updating policy of the conflict list as “first save first leave”.

Fig. 1.

Structure and updating direction of conflict list.

Conflict flag is designed to trig the combination of the objectives when conflict list has the same objectives and the minimum conflict of these objectives are smaller than a predefined conflict threshold C_th. Conflict flag is set which represents that non-conflict or weak-conflict objectives are found and the optimization model can be updated. Conflict flag can be decided with: $\begin{eqnarray}\mathit{Conflict}\_\mathit{flag}=\left\{\begin{array}{@{}ll@{}}1\quad & C_{\text{min}}<∼C_{th}\\ 0\quad & \text{else}\end{array}\right.\end{eqnarray}$ (5)

2.2. Modified Two-Arch 2 for objective reduction

In Two-Arch 2, DA and CA are used to balance the exploitation and exploration. DA is updated to enhance the diversity of the population and CA is used to speed up the convergence of the algorithm. Due to the different functions of DA and CA, different selection principles are assigned to the two archives. In DA, Pareto based selection mechanism is adopted; while in CA indicator I_ϵ+ from IBEA [6] is used, which is defined as [8]: $\begin{eqnarray}I_{{\varepsilon}+}(x_{1},x_{2})=\min _{{\varepsilon}}(f_{i}(x_{1})-{\varepsilon}\leq f_{i}(x_{2}),1\leq i\leq m)\end{eqnarray}$ (6) Where, m is the number of objectives. As shown in (6), I_ϵ+ is an indicator which demonstrates the minimum distance that x₁ needs to dominate x₂. The individuals with smaller I_ϵ+ have more chance to remove from CA because they are close to the dominated solutions at least in one dimension in the objective space. I_ϵ+ is an effective tool to increase the evolutional pressure in high dimensional problem. However, to obtain I_ϵ+, calculations need to be done between two individuals, which increases the computational burden. Therefore, the evolutionary procedure in original Two-Arch 2 should be modified to avoid excessive calculations. Because there are two possible cases after the redundant objectives are found and processed by the interactive aggregation trees frame. One is the number of updated objectives are less than four, referred as case 1; the other one is that the number of updated objectives are still more than four, referred as case 2. In case 1, non-dominated relationship can produce enough evolutionary pressure to push the algorithm moving towards Pareto front. Therefore, only DA is used in the evolutionary process to compute non-dominated solutions; CA is not used in case 1 to decrease the computational burden on calculating the indicator I_ϵ+. In case 2, it is still a high dimensional problem and both CA and DA are used to obtain the Pareto front.

2.3. Algorithm description

To demonstrate the proposed algorithm intuitionally, the flow chart of proposed algorithm is shown in Fig. 2. In Fig. 2, the operations in the first broken-line box are interactive aggregation trees frame. At the beginning of the algorithm, original sampling points are generated randomly, which may not well distribute over the objective space. The conflict degree based on these sampling points cannot fully represent the real conflict relationship of objectives. As the evolution proceeding, the sampling points are evolved both in Pareto rank and distribution. When the recorded data in conflict list is relatively stable and conflict flag is set, it is considered that sampling points are uniformly distributed in the objective space and the redundant objectives can be identified. The operations in the second broken-line box are major modifications in evolutionary process of Two-Arch2. M_C is the objective number of current model, which determines the archives to be used in the evolutionary mechanism. If M_C is smaller than 4, the high dimensional optimization problem degrades to a traditional multi-objective problem and non-domination comparison in DA is eligible for the evolution.

Fig. 2.

Flow chart of the proposed algorithm.

To facilitate the implementation of the proposed algorithm, the details about its iterative process are given as:

Step 1: Initialize the function evaluation number t, conflict_flag, conflict_list, size of DA and CA. Define the maximum function evaluation number FE_max, the conflict threshold C_th, the number of objectives M.

Step 2: Generate N sampling points in the feasible design space; calculate their objective function values and update function evaluation number t.

Step 3: Compute the conflict degree of objectives with (1)–(4); update the conflict_list and conflict_flag. If conflict_flag is set, update the optimization model by combining the objectives based on the aggregation trees and update M_C.

Step 4: If the current number of objectives M_C < 4, update only DA and select parents from DA. If M_C ≥ 4, update both DA and CA; Select parents from DA and CA.

Step 5: If t > FE_max, go to step 6; else generate the offspring from parents and calculate their objective function values and update function evaluation number t. If the model has updated based on the interactive aggregation trees frame, go to step 4; else go to step 3.

Step 6: Stop the algorithm.

3. Numerical validations

To validate and demonstrate advantages of the proposed algorithm, it is used to solve DTLZ test functions [9] and extended TEAM Workshop Problem 22 [10].

Table 1
Definition of DTLZ2 and DTLZ5

Name Problems Parameter domain

DTLZ2 $\begin{array}{@{}l@{}}\displaystyle f_{1}=(1+g)\prod _{i=1}^{M-1}\cos (y_{i}{\pi}/2)\\[10.0pt] \displaystyle f_{2}=(1+g)\prod _{i=1}^{M-2}\cos (y_{i}{\pi}/2)\sin (y_{M-2+1}{\pi}/2)\\[5.0pt] \ldots \\[5.0pt] \displaystyle f_{M}=(1+g)\sin (y_{1}{\pi}/2)\\[5.0pt] \displaystyle g=\sum _{i=1}^{k}(z_{i}-0.5)^{2}\end{array}$ 0 ≤ x_i ≤ 1

DTLZ5 As DTLZ2, except all y₂…y_M−1 ∈ x are replaced by (1 + 2gy_i)∕2(1 + g) 0 ≤ x_i ≤ 1

Name	Problems	Parameter domain
DTLZ2	$\begin{array}{@{}l@{}}\displaystyle f_{1}=(1+g)\prod _{i=1}^{M-1}\cos (y_{i}{\pi}/2)\\[10.0pt] \displaystyle f_{2}=(1+g)\prod _{i=1}^{M-2}\cos (y_{i}{\pi}/2)\sin (y_{M-2+1}{\pi}/2)\\[5.0pt] \ldots \\[5.0pt] \displaystyle f_{M}=(1+g)\sin (y_{1}{\pi}/2)\\[5.0pt] \displaystyle g=\sum _{i=1}^{k}(z_{i}-0.5)^{2}\end{array}$	0 ≤ x_i ≤ 1

DTLZ5	As DTLZ2, except all y₂…y_M−1 ∈ x are replaced by (1 + 2gy_i)∕2(1 + g)	0 ≤ x_i ≤ 1

3.1. DTLZ test functions

The proposed algorithm is firstly tested on two typical test functions: DTLZ2 and DTLZ5 with 20 objectives and 20 variables as shown in Table 1. The objective of DTLZ2 as being separable irrespective, as attempting to optimize them one parameter at a time will identify at least one global optima; DTLZ5 is problem with degenerated Pareto front. The main parameters of the proposed algorithm to solve DTLZ2 and DTLZ5 are set as: N = 100, FE_max = 20000, C_th = 0.3, M = 20, CAsize = 100, DAsize = 100.

Inverted generation distance (IGD) represents distance between the searched solutions by algorithms and the real Pareto solutions. The smaller the IGD is, the closer the obtained solutions to the real Pareto solutions are. Table 2 is the comparison on the smallest, largest, mean and standard deviation (std) of IGD values among NSGAIII [11], Two-Arch2 [1], IBEA [8] and the proposed method solving DTLZ2 and DTLZ5 in 30 random runs. From Table 2, it is shown that the IGD index of the proposed algorithm in DTLZ2 ranks in the second or third places, which demonstrates that the proposed method can find competitive solutions in problems with conflicting objectives; the proposed method has the best performance on IGD index in DTLZ5, which addresses the advantages of the proposed method in solving the problems containing redundant objectives. With the frame of aggregation trees, 6 weak-conflict objectives are combined into 2 compound objectives in DTLZ2 and 16 weak-conflict objectives are combined into 2 compound objectives in DTLZ5. Figure 3 is the aggregation trees of DTLZ5 obtained by the proposed method. As shown in Fig. 3, f₁ and f₂, f₃ and f₄, f₇ and f₈, f₁₀ and f₁₁, f₁₇ and f₁₈ are the five pairs of objectives whose conflict degrees are under the conflict threshold in the first round. The aggregation operation is continued until all the conflict degrees of current objectives are higher than the conflict threshold.

Table 2
IGD comparison between the proposed method and NSGAIII, Two-Arch2 and IBEA

Minimum Maximum Mean Std

DTLZ2 DTLZ5 DTLZ2 DTLZ5 DTLZ2 DTLZ5 DTLZ2 DTLZ5

NSGAIII 0.9226 0.1001 1.0487 0.3058 0.9664 0.1781 0.0263 0.0530

Two-Arch2 0.7304 0.0840 0.7661 0.2252 0.7498 0.1436 0.0084 0.0359

IBEA 0.6481 0.0078 0.6766 0.3028 0.6597 0.1424 0.0073 0.0747

Propose method 0.7147 0.0010 0.7834 0.0753 0.7480 0.0273 0.0192 0.0161

	Minimum	Maximum	Mean	Std
NSGAIII	0.9226	0.1001	1.0487	0.3058	0.9664	0.1781	0.0263	0.0530
Two-Arch2	0.7304	0.0840	0.7661	0.2252	0.7498	0.1436	0.0084	0.0359
IBEA	0.6481	0.0078	0.6766	0.3028	0.6597	0.1424	0.0073	0.0747
Propose method	0.7147	0.0010	0.7834	0.0753	0.7480	0.0273	0.0192	0.0161

Fig. 3.

Aggregation trees of DTLZ5.

3.2. Extended TEAM workshop Problem 22

To comprehensively explore performances of the proposed algorithm, it is used to optimize the extended TEAM Workshop Problem 22. The problem is a superconducting magnetic energy storage system (SMES) design optimization, as shown in Fig. 4. The system is consisted of an inner main solenoid and an outer shielding solenoid. The design of SMES is to obtain the desired store energy with minimal stray fields. The magnetic flux of stray filed is calculated with the average magnetic flux density of 22 equidistance points along line A and line B in the original Problem 22. In our case, magnetic flux density on each point is computed as one objective in order to have precise information about stray field. The optimal problem of extended Problem 22 is formulated as: $\begin{eqnarray}\begin{array}{@{}l@{}}\min \quad \left\{\begin{array}{@{}l@{}}F_{i}=B_{\text{stay},i}^{2}/B_{\text{nor}}^{2}\quad (i=1,2,\ldots ,22)\quad \\[5.0pt] F_{23}=|E-E_{\text{ref}}|/E_{\text{ref}}\quad \end{array}\right.\\[6.0pt] s.t.\quad |B_{\text{max}}|\leq 4.92∼T\end{array}\end{eqnarray}$ (7) where: E_ref = 180 MJ, B_nom = 3 × 10⁻³ T, Energy is the practical energy storage, B_max is the maximum magnetic flux density, $B_{\text{stray},i}^{2}$ is calculated in 22 equidistance points along line A and line B.

Fig. 4.

The coils structure of SMES.

Fig. 5.

Comparison of minimum objective values.

In this case study, the main parameters of the proposed algorithm to solve Problem 22 are set as: N = 100, FE_max = 20000, C_th = 0.3, M = 23, CAsize = 100, DAsize = 100. Figure 5 is the comparison of minimum values on each objective searched by NSGAIII, IBEA, Two Arch2 and the proposed method. To make a fair comparison, the stopping criteria are the same in each algorithm, that is when the maximum objective function calls reach 20000, the algorithm is terminated. From Fig. 5, it is shown that the variation tendency of the minimum value on each objective function is the same which demonstrates the presented algorithms can converge towards the Pareto front. Two Arch2 and the proposed method have searched smaller objective values on most of the objectives. 12 minimum objective values found by the proposed method are the smallest, which indicates the proposed method has a strong searching ability.

Table 3 is the comparison of solutions and computational complexity. Dominance relationships are calculated among the obtained solutions by NSGAIII, IBEA, Two-Arch2, the proposed method and a benchmark solution in a single objective optimization of Problem 22 [12]. Dominance relationships can indicate the relative quality between solutions and the number of solutions dominated by other solutions can demonstrate the searching ability of algorithms over the entire objective space. From Table 3, no solutions searched by the proposed method are dominated by the benchmark solution; 5 of 100 solutions found by the proposed method are dominated by other solutions. In the aspect of computational complexity, m is number of objectives; N is number of population; m′ is the number of objectives in proposed method. Because of the objective reduction frame, m′≤ m. The proposed method can decrease the computational complexity of the algorithm.

Table 3

Comparison on solutions and complexity of Problem 22

	No. of dominated solutions by benchmark	No. of dominated solutions overall	Computational complexity
NSGAIII	0	20	$\max \{O(N\text{log}^{m-2}N),O(mN^{2})\}$
IBEA	2	2	O (mN²)
Two-Arch2	0	7	$\max \{O(N\text{log}^{m-2}N),O(mN^{2})\}$
The proposed method	0	5	$\max \{O(N\text{log}^{m^{\prime }-2}N),O(m^{\prime }N^{2})\}$

Table 4 is the solution comparison between benchmark solution and a solution obtained by the proposed method. It is shown that the proposed method can find a solution which is better than benchmark solution in the aspect of energy storage. It is shown that the proposed method can provide more options for the designer.

Fig. 6.

Aggregation trees of extended Problem 22.

Table 4

Solution comparison between benchmark solution and solution obtained by the proposed method

	r₂ (m)	h₂ ∕2(m)	d₂ (m)	Min F_i	Max F_i	F ₂₃
Benchmark solution	3.080	0.254	0.370	2.100 × 10⁻⁷	0.214	0.00556
The proposed method	3.091	0.344	0.282	9.678 × 10⁻¹⁰	0.397	0.00111

Figure 6 is the aggregation trees of Problem 22. As shown in Fig. 6, the conflict degrees of 4 groups are below the conflict threshold; the objectives of magnetic flux density in adjacent points have smaller conflict degrees, which are in accord with the real magnetic field distribution; energy storage is most conflict objective to the stray field with which the conflict degree goes to 87.16%. After the aggregation trees operation, 23 objectives can be decreased to 5.

4. Conclusions

A Two-arch2 method with aggregation trees frame for solving high dimensional optimization problems is introduced in this paper. The numerical results on the DTLZ test functions and the Problem 22 demonstrate that the proposed algorithm has the ability to find the weak-conflict objectives and converge to a satisfied quality Pareto front with combined objectives. The proposed method decreases complexity of the algorithm compared by Two-arch 2 by reducing the number of objectives and aggregation trees method also provide a visual image of conflict degree for the decision maker. In electromagnetic optimization problem, numerical calculation is the dominate factor to time duration. The number of objective function calls is widely used to represent the time consumption of electromagnetic optimization. As the computational burden is still the main obstacle in the electromagnetic design, the authors will make endeavors to decrease the objective function calls in the future research.

Footnotes

Acknowledgements

This work was supported by the Natural Science Foundation, China under Grant No. 52077203, Natural Science Foundation of Zhejiang Province No. LZ22F040002 and the Fundamental Research Funds for the Provincial Universities of Zhejiang 2021YW01.

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