An improvement of clustering-based objective reduction method for many-objective optimization problems

ORc
 LPCA 1.00 0.01 0.00 0.06 LPCA 0 1 1 0
 OC-ORA 0.01 1.00 0.17 0.12 OC-ORA 1 0 0 0
 CORi 0.00 0.17 1.00 0.01 CORi 1 0 0 1
 CORc 0.06 0.12 0.01 1.00 CORc 0 0 1 0
level at 0.05, and the alternative hypothesis is that the performance of the two methods
is significant different. The Table 2a shows p values when comparing two algorithms
with each other. The Table 2b presents the hypotheses are accepted or rejected. The
value 0 means that the null hypothesis is accepted, in other words, the two algorithms
are similar with significant level at 0.05. The value 1 means that the null hypothesis
is rejected or we accept the alternative hypothesis which mean that the two algorithms
are different with significant level at 0.05. As can be seen from Table 2b, the LPCA is
different to OC-ORA, or CORi but similar to CORc at significant level 0.05. CORi is
similar to OC-ORA but different to LPCA, and CORc.
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Table 3. The average number of times objective reduction algorithms calling the MOEAs when it finds
 the set of conflicting objectives
 DTLZ5IM objective reductions
 MaOEAs
 I M LPCA OC-ORA CORi CORc
 2 5 2.0 3.8 2.0 2.0
 3 5 2.0 3.0 2.0 2.0
 GrEA 5 10 2.0 6.0 2.0 2.0
 7 10 2.0 4.0 2.0 2.0
 5 20 1.2 15.7 2.1 2.1
 2 5 2.0 3.8 2.0 2.0
 3 5 2.0 3.1 2.3 2.0
 KnEA 5 10 2.0 6.0 2.3 2.0
 7 10 2.0 4.1 2.4 2.0
 5 20 2.0 16.0 2.1 2.6
 2 5 2.0 4.0 2.0 2.0
 3 5 2.0 2.9 2.0 2.0
 NSGAIII 5 10 2.0 6.0 2.0 2.0
 7 10 2.0 4.0 2.0 2.0
 5 20 2.2 15.9 2.5 3.1
 2 5 2.0 3.8 2.0 2.0
 3 5 2.0 3.0 2.0 2.0
 RVEA 5 10 2.0 6.0 2.0 2.0
 7 10 2.0 4.0 2.1 2.0
 5 20 2.0 17.4 2.2 2.1
 2 5 2.0 4.0 2.0 2.0
 3 5 2.0 2.9 2.0 2.0
 θ-DEA 5 10 2.0 6.0 2.0 2.0
 7 10 2.0 4.0 2.1 2.0
 5 20 2.0 16.5 2.1 2.6
 2 5 2.0 4.0 2.1 2.0
 3 5 2.0 3.0 2.1 2.0
 LMEA 5 10 2.0 6.0 2.1 2.0
 7 10 2.0 4.0 2.1 2.0
 5 20 2.3 16.0 2.1 2.3
 Beside the number of times successful in finding non-redundant objective set above,
the number of repeating in reduction process is considered when do comparison the
performance of the objective reduction algorithm. For OC-ORA algorithm, each loop
can reduce at most one redundant objective. Table 3 shows that COR and LPCA can
execute almost 2 loops to finish the process, while OC-ORA need M − I + 1 loops.
As a result, COR and LPCA are equivalent, and they more efficient than the OC-ORA,
specially in high redundant instances.
6. Conclusion and Future Works
 The paper proposed a clustering-based objective reduction method which uses PAM
algorithm for clustering objectives in MaOPs for determining redundant objectives.
The proposed algorithm used Silhouette index to self-determine the number of clusters
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 Journal of Science and Technique - Le Quy Don Technical University - No. 202 (10-2019)
instead requiring it as a parameter. We compared the proposed method with two other
existing methods on different instances of DTLZ5IM(I,M) problem. The result showed
that the proposed algorithm is not only equivalent to or more efficient than existing
methods but also it can retain comparable results as existing methods can do.
 There are other methods which are able to automatically determine the number of
clusters, future works could investigate these methods for the proposed method.
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Journal of Science and Technique - Le Quy Don Technical University - No. 202 (10-2019)
 Nguyen Xuan Hung received his B.Sc (1998) and M.Sc (2008) degrees in computer science
 and technology from Le Quy Don Technical University. Currently he is a PhD Candidate at
 the Faculty of Information Technology, Le Quy Don Technical University. His current research
 interests include Evolutionary Multi-Objective/Many-Objective Optimization and Objective
 Dimensionality Reduction. His email is nguyenxuanhung@outlook.com
 Bui Thu Lam received the Ph.D. degree in computer science from the University of New
 South Wales (UNSW), Australia, in 2007. He did postdoctoral training at UNSW from 2007
 until 2009. He has been involved with academics including teaching and research since 1998.
 Currently, he is an Associate Professor at Le Quy Don Technical University, Hanoi, Vietnam.
 He is doing research in the field of evolutionary computation, specialized with evolutionary
 multi-objective optimization.
 Tran Cao Truong received the B.Sc in applied mathematics and informatics from from Ha
 Noi University of Science, Vietnam, in 2005 and the M.S. degree from Le Qui Don Technical
 University, Viet Nam in 2009. He received the PhD degree in computer science from Victoria
 University of Wellington, New Zealand, in 2018. He is researching in the field of evolutionary
 computation, specialized with evolutionary machine learning for data mining with missing data.
 He servers as a reviewer of international journals, including IEEE Transactions on Evolution-
 ary Computation, IEEE Transactions on Cybernetics, Pattern Recognition, Knowledge-Based
 Systems, Applied Soft Computing and Engineering Application of Artificial Intelligence. He
 is also a PC member of international conferences, including IEEE Congress on Evolutionary
 Computation, IEEE Symposium Series on Computational Intelligence, the Australasian Joint
 Conference on Artificial Intelligence and the AAAI Conference on Artificial Intelligence.
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Section on Information and Communication Technology (ICT) - No. 14 (10-2019)
 GIẢI THUẬT GIẢM CHIỀU MỤC TIÊU CẢI TIẾN
 SỬ DỤNG PHƯƠNG PHÁP PHÂN CỤM
 CHO BÀI TOÁN TỐI ƯU NHIỀU MỤC TIÊU
 Tóm tắt
 Bài toán tối ưu nhiều mục tiêu đã nhận được sự quan tâm đáng kể từ các nhà nghiên
 cứu gần đây. Khi số lượng mục tiêu tăng lên, chúng gây ra những khó khăn cho các thuật
 toán tiến hóa đa mục tiêu hiện tại về toán tử lựa chọn, yêu cầu kích thước quần thể lớn, việc
 trực quan hóa tập kết quả, đặc biệt là chi phí tính toán. Một số bài toán có mục tiêu dư thừa
 và có thể bị thoái hóa thành các bài toán có kích thước thấp hơn. Việc loại bỏ các mục tiêu
 dư thừa sẽ tiết kiệm thời gian và không gian khi giải quyết bài toán. Bài báo này cải tiến
 phương pháp giảm kích thước sử dụng thuật toán PAM thực hiện phân cụm các mục tiêu sau
 đó loại bỏ các mục tiêu dư thừa. Hai biến thể của thuật toán đề xuất tự xác định số mục tiêu
 dư thừa và hiệu quả hơn thuật toán gốc.
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