Ranking Constraints via Topological Dual-Directional Search in Evolutionary Multi-Objective Optimization
Authors: Ruiqing Sun, Dawei Feng, Sheng Qi, Xing Zhou, Lianghao Li, Bo Ding, Yijie Wang, Rui Wang, Huaimin Wang
Organizations: National Key Laboratory of Parallel and Distributed Computing, College of Computer Science and Technology, National University of Defense Technology, Changsha 410000, P.R. China · College of System Engineering, National University of Defense Technology, Changsha 410000, P.R. China · College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, P.R. China · State Key Laboratory of Complex & Critical Software Environment, College of Information and Communication, National University of Defense Technology, Wuhan 430019, P.R. China
Existing evolutionary algorithms for Constrained Multi-objective Optimization Problems (CMOPs) typically treat all constraints uniformly, overlooking their distinct geometric relationships with the true Constrained Pareto Front (CPF). In reality, constraints play different roles: some directly shape the final CPF, some create infeasible obstacles, while others are irrelevant. To exploit this insight, we propose a novel algorithm named RCCMO, which sequentially performs unconstrained exploration, single-constraint exploitation, and full-constraint refinement. The core innovation of RCCMO lies in a constraint prioritization method derived from these geometric insights, seamlessly coupled with a unique dual-directional search mechanism. Specifically, RCCMO first prioritizes constraints that constitute the final CPF, approaching them from the evolutionary direction (optimizing objectives) to locate the CPF directly shaped by single-constraint boundaries. Subsequently, for constraints that merely hinder the population's progress, RCCMO searches from the anti-evolutionary direction (targeting the infeasible boundaries where hindering constraints intersect with the CPF) to effectively discover how these constraints obstruct and form the final CPF. Meanwhile, irrelevant constraints are intentionally bypassed. Furthermore, a series of specialized mechanisms are proposed to accelerate the algorithm's execution, reduce heuristic misjudgments, and dynamically adjust search directions in real time. Extensive experiments on 5 benchmark test suites and 29 real-world CMOPs demonstrate that RCCMO significantly outperforms seven state-of-the-art algorithms.
Figures & tables
Figure 1: Three types of constraint priority situations based on their geometric relationships with the final CPF.
Figure 2: Flowchart of the proposed RCCMO
Figure 3: Working principle of Probe population: used to detect constraints that hinder the main population from further evolving towards the Pareto optimal direction.
Figure 4: A step-by-step example demonstrating the constraint prioritizing mechanism in RCCMO.
Problem Suite
APSEA
C3M
FCDS
IMTCMO
MCCMO
MSCMO
MTOTC
RCCMO
DASCMOP ( + / − / ≈ )
4/5/0
1/6/2
5/3/1
3/3/3
0/6/3
0/9/0
0/7/2
-
Avg. Rank
5.22
5.11
2.67
3.67
4.89
6.22
5.11
3.11
DOC ( + / − / ≈ )
0/8/1
3/6/0
0/7/2
3/2/4
4/4/1
0/8/1
0/6/3
-
Avg. Rank
6.22
3.67
6.22
2.67
2.44
7.11
5.22
2.44
LIRCMOP ( + / − / ≈ )
0/14/0
0/14/0
2/9/3
1/11/2
1/12/1
0/13/1
1/9/4
-
Avg. Rank
7.93
5.21
3.29
4.93
3.71
5.50
3.86
1.57
Table 1: Statistical Summary of IGD Values ( + / − / ≈ ) and Average Ranks across Test Suites.
Problem Suite
APSEA
C3M
FCDSU
IMTCMO
MCCMO
MSCMO
MTOTC
RCCMO
RWMOP ( + / − / ≈ )
5/15/8
5/15/8
5/17/6
6/10/12
7/15/6
2/20/6
8/13/7
-
Avg. Rank
4.61
4.46
5.32
4.18
4.43
5.93
3.86
3.21
Note: Nemenyi test Critical Difference (CD) = 1.98 at α=0.05 .
Table 2: Statistical Summary of HV Values ( + / − / ≈ ) and Average Ranks across RWMOPs.
Problem Suite
APSEA
C3M
FCDS
IMTCMO
MCCMO
MSCMO
MTOTC
RCCMO w/o AUS
RCCMO
DASCMOP
7.74 (1.32)
29.12 (7.73)
43.68 (2.99)
14.66 (6.63)
45.20 (29.62)
13.14 (5.33)
64.59 (18.79)
22.05 (5.67)
16.62 (5.33)
DOC
14.04 (3.94)
35.69 (30.37)
46.02 (16.95)
17.07 (5.43)
25.15 (10.98)
33.04 (10.47)
70.96 (36.81)
25.51 (4.07)
25.60 (4.84)
LIRCMOP
9.54 (4.10)
13.25 (4.47)
61.60 (18.83)
21.84 (11.42)
22.02 (11.12)
12.22 (3.79)
45.09 (35.91)
19.62 (2.23)
19.15 (2.22)
SDC
11.76 (3.14)
13.75 (4.32)
39.00 (12.30)
13.66 (3.64)
20.93 (5.63)
13.57 (11.77)
29.21 (16.41)
20.02 (2.88)
19.66 (3.56)
ZXH_CF
14.98 (4.85)
19.46 (6.83)
73.25 (13.64)
24.69 (6.19)
33.76 (13.95)
14.36 (4.10)
92.00 (30.72)
24.31 (5.58)
23.31 (6.44)
Overall Average
11.61
22.25
52.71
18.38
29.41
17.26
60.37
22.30
20.87
Table 3: Average Running Time (in seconds) and Standard Deviation across Test Suites. The shortest time is bolded , and the second shortest is underlined .
Figure 5: Comparison of the performance and running time of RCCMO under different interval parameter V .
Problem Suite
w/o AUS
Only-Neg
w/o Re-rank
w/o Flip
Only-Pos
Rand-Priority
RCCMO (Baseline)
DASCMOP ( + / − / ≈ )
0/0/9
0/0/9
1/2/6
0/0/9
0/2/7
1/3/5
-
Avg. Rank
3.44
3.67
4.56
4.56
3.44
4.22
4.11
DOC ( + / − / ≈ )
1/0/8
1/1/7
0/2/7
0/1/8
0/6/3
0/2/7
-
Avg. Rank
3.33
3.67
3.56
4.00
6.33
4.33
2.78
LIRCMOP ( + / − / ≈ )
0/1/13
0/8/6
0/1/13
1/1/12
2/4/8
1/2/11
-
Avg. Rank
4.07
5.50
3.64
4.14
3.86
3.50
3.29
Table 4: Statistical Summary of IGD Values ( + / − / ≈ ) and Average Ranks for Ablation Variants across Test Suites.
College of Computer Science and Technology, National University of Defense Technology, Changsha 410000, P.R. China · College of Intelligence Science and Technology, National University of Defense Technology, Changsha, 410000, P.R. China · State Key Laboratory of Complex & Critical Software Environment, College of Information and Communication, National University of Defense Technology, Wuhan 430019, P.R. China +1
Department of Engineering, University of Toyama, Toyama-shi 930-8555, Japan · Cyberscience Center, Tohoku University, Sendai-shi 980-8578, Japan · Faculty of Science and Technology, Hirosaki University, Hirosaki-shi 036-8560, Japan +5
Department of Mechanical Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada · Department of Civil Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada