RDEx-CMOP: Feasibility-Aware Indicator-Guided Differential Evolution for Fixed-Budget Constrained Multiobjective Optimization

Sichen Tao; Yifei Yang; Ruihan Zhao; Kaiyu Wang; Sicheng Liu; Shangce Gao

cs.NEApr 4, 2026

RDEx-CMOP: Feasibility-Aware Indicator-Guided Differential Evolution for Fixed-Budget Constrained Multiobjective Optimization

Authors: Sichen Tao, Yifei Yang, Ruihan Zhao, Kaiyu Wang, Sicheng Liu, Shangce Gao

Organizations: 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 · Sino-German College of Applied Sciences, Tongji University, Shanghai 200092, China · School of Mechanical Engineering, Tongji University, Shanghai 200092, China · Chongqing Institute of Microelectronics Industry Technology, University of Electronic Science and Technology of China, Chongqing 401332, China · Artificial Intelligence and Big Data College, Chongqing Polytechnic University of Electronic Technology, Chongqing 401331, China · Department of Information Engineering, Yantai Vocational College, Yantai 264670, China

Abstract

Constrained multiobjective optimisation requires fast feasibility attainment together with stable convergence and diversity preservation under strict evaluation budgets. This report documents RDEx-CMOP, the differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session) constrained multiobjective track. RDEx-CMOP integrates an ε-level feasibility schedule, a SPEA2-style indicator-driven fitness assignment, and a fitness-oriented current-to-pbest/1 mutation operator. We evaluate RDEx-CMOP on the official CEC 2025 CMOP benchmark using the median-target U-score framework and the released trace data. Experimental results show that RDEx-CMOP achieves the highest total score and the best overall average rank among all released comparison algorithms, with strong target-attainment behaviour and near-zero final violation on most problems.

Figures & tables

Rank	Algorithm	Total Score	Avg Score/Prob.	Speed	Accuracy	Constraint
1	RDEx-CMOP	58456.0	3897.07	52875.5	5499.5	81.0
2	DESDE	51571.0	3438.07	45171.0	6290.0	110.0
3	CCEMT	41902.5	2793.50	30374.5	11042.0	486.0
4	IMTCMO	40038.0	2669.20	26186.5	13582.5	269.0
5	CCPTEA	39232.0	2615.47	24948.0	14138.0	146.0
6	MTCMMO	13302.5	886.83	2996.5	10069.0	237.0

TABLE I: CEC 2025 CMOP evaluation (median target): overall scores over 15 problems and 30 runs for all released comparison algorithms.

Competitor	Final Q			TTT
Competitor	W/T/L	Holm	$A_{12}$	W/T/L	Holm	$A_{12}$
CCEMT	12/0/3	12/0/3	0.82	11/1/3	11/1/3	0.87
CCPTEA	12/2/1	12/2/1	0.80	11/3/1	11/3/1	0.82
DESDE	11/3/1	8/6/1	0.71	8/5/2	6/7/2	0.68
IMTCMO	12/1/2	11/2/2	0.81	11/1/3	9/4/2	0.80
MTCMMO	15/0/0	15/0/0	1.00	13/2/0	13/2/0	0.93

TABLE II: Primary pairwise summary over the 15 CEC2025 CMOP functions (30 runs). We report uncorrected per-function Wilcoxon W/T/L at

\alpha=0.05

, Holm-corrected W/T/L across functions, and the median Vargha–Delaney

A_{12}

effect size for feasibility-aware final quality and time-to-target (larger is better for minimization).

Algorithm	Final Q	TTT
RDEx-CMOP	1.77	1.80
CCEMT	3.33	3.47
CCPTEA	3.80	4.17
DESDE	2.50	2.50
IMTCMO	3.73	4.03
MTCMMO	5.87	5.03

TABLE III: Primary Friedman tests on per-function medians over the 15 CEC2025 CMOP functions (30 runs). Final Q:

\chi^{2}=41.90

df=5

p=1.68E-07

; TTT:

\chi^{2}=29.88

df=5

p=2.30E-05

. Lower average rank indicates better performance.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Rank	Algorithm	Total Rank	Avg Rank/Prob.	Avg Speed	Avg Accuracy	Avg Constraint
1	RDEx-CMOP	25.0	1.67	1.77	5.00	3.73
2	DESDE	39.0	2.60	2.60	4.33	3.67
3	CCEMT	47.0	3.13	3.13	3.40	3.40
4	IMTCMO	54.0	3.60	3.80	2.67	3.23
5	CCPTEA	60.0	4.00	3.80	2.20	3.60
6	MTCMMO	90.0	6.00	5.90	3.40	3.37

Appendix

TABLE IV: CEC 2025 CMOP evaluation (median target): average rankings over 15 problems (lower is better) for all released comparison algorithms.

Problem

RDEx-CMOP

CCEMT

CCPTEA

DESDE

IMTCMO

MTCMMO

Mean

4.03E-01

1.86E-01

3.15E-01

6.25E-01

3.60E-01

2.21E-01

2.15E-01

2.00E-01

4.31E-01

6.11E-01

7.60E+00

2.00E+00

2.15E-02

1.03E-02

3.20E-02

1.13E-02

3.55E-02

1.31E-02

2.56E-02

7.65E-03

3.48E-02

1.15E-02

4.62E-01

1.57E-01

9.48E-01

1.17E-01

1.06E+00

4.56E-02

1.09E+00

5.27E-02

9.98E-01

1.25E-01

1.08E+00

3.36E-02

1.18E+00

1.73E-01

1.01E+00

1.44E-01

1.17E+00

1.38E-01

1.11E+00

6.54E-02

1.09E+00

2.67E-02

1.11E+00

4.98E-02

5.61E+02

3.09E+02

4.62E+01

2.29E+01

9.64E+01

4.11E+01

9.95E+01

3.74E+01

6.50E+01

2.88E+01

1.11E+02

4.51E+01

1.83E+02

7.11E+01

Appendix

TABLE V: Feasibility-aware final-quality comparison on the 15 CEC2025 CMOP functions. For each run, the final-quality score equals the final objective/IGD value for feasible runs and

B_{p}+\mathrm{CV}

for infeasible runs, where

B_{p}

is the largest finite final objective/IGD value on problem

p

plus

1

(smaller is better).

Problem	RDEx-CMOP		CCEMT			CCPTEA			DESDE			IMTCMO			MTCMMO
Problem	Mean	SD	Mean	SD	W	Mean	SD	W	Mean	SD	W	Mean	SD	W	Mean	SD	W
1	884.4	233.7	721.8	183.4	-	864.0	139.2	=	634.9	247.7	-	833.9	164.3	-	1001.0	0.0	=
2	662.0	232.4	927.5	92.4	+	936.6	102.5	+	792.8	149.0	+	943.9	89.4	+	1001.0	0.0	+
3	635.1	79.5	968.4	42.0	+	991.1	29.2	+	740.6	147.9	+	990.9	25.9	+	948.1	139.5	+
4	708.2	139.0	897.0	175.6	+	844.5	201.8	+	816.6	128.9	+	819.3	222.7	+	1001.0	0.0	+
5	399.0	256.5	823.2	212.5	+	874.3	211.6	+	490.9	277.6	=	893.0	176.6	+	966.3	134.1	+

Appendix

TABLE VI: Time-to-target comparison on the 15 CEC2025 CMOP functions. TTT is the first checkpoint index where the run reaches the median target (smaller is better); runs that never reach the target are assigned 1001.

Competitor	Final IGD			Final CV
Competitor	W/T/L	Holm	$A_{12}$	W/T/L	Holm	$A_{12}$
CCEMT	12/0/3	12/0/3	0.82	8/6/1	8/6/1	0.77
CCPTEA	12/2/1	12/2/1	0.80	15/0/0	15/0/0	1.00
DESDE	11/3/1	8/6/1	0.71	15/0/0	15/0/0	1.00
IMTCMO	12/1/2	11/2/2	0.81	15/0/0	15/0/0	1.00
MTCMMO	15/0/0	15/0/0	1.00	15/0/0	15/0/0	1.00

Appendix

TABLE VII: Complementary pairwise summary over the 15 CEC2025 CMOP functions (30 runs). For each metric (Final IGD and Final CV), we report uncorrected per-function Wilcoxon W/T/L at

\alpha=0.05

, Holm-corrected W/T/L across functions, and the median Vargha–Delaney

A_{12}

effect size (larger is better for minimization).

Algorithm	Final IGD	Final CV
RDEx-CMOP	1.77	1.30
CCEMT	3.13	1.70
CCPTEA	4.00	4.50
DESDE	2.50	4.50
IMTCMO	3.80	4.50
MTCMMO	5.80	4.50

Appendix

TABLE VIII: Complementary Friedman tests on per-function medians over the 15 CEC2025 CMOP functions (30 runs). Final IGD:

\chi^{2}=41.87

df=5

p=1.70E-07

; Final CV:

\chi^{2}=51.77

df=5

p=3.20E-09

. Lower average rank indicates better performance for each metric.

Problem

RDEx-CMOP

CCEMT

CCPTEA

DESDE

IMTCMO

MTCMMO

Mean

4.03E-01

1.86E-01

3.15E-01

6.25E-01

3.60E-01

2.21E-01

2.15E-01

2.00E-01

4.31E-01

6.11E-01

7.60E+00

2.00E+00

2.15E-02

1.03E-02

3.20E-02

1.13E-02

3.55E-02

1.31E-02

2.56E-02

7.65E-03

3.48E-02

1.15E-02

4.62E-01

1.57E-01

9.48E-01

1.17E-01

1.06E+00

4.56E-02

1.09E+00

5.27E-02

9.98E-01

1.25E-01

1.08E+00

3.36E-02

1.18E+00

1.73E-01

1.01E+00

1.44E-01

1.17E+00

1.38E-01

1.11E+00

6.54E-02

1.09E+00

2.67E-02

1.11E+00

4.98E-02

5.61E+02

3.09E+02

4.62E+01

2.29E+01

9.64E+01

4.11E+01

9.95E+01

3.74E+01

6.50E+01

2.88E+01

1.11E+02

4.51E+01

1.83E+02

7.11E+01

Appendix

TABLE IX: Final IGD comparison on the 15 CEC2025 CMOP functions. For each algorithm, the mean and SD over 30 runs are reported;

W

gives the Wilcoxon outcome of RDEx-CMOP against the competitor.

Problem

RDEx-CMOP

CCEMT

CCPTEA

DESDE

IMTCMO

MTCMMO

Mean

-3.19E-01

7.94E-03

-2.43E-01

4.35E-02

0.00E+00

-4.27E-01

1.70E-02

-4.51E-01

7.95E-02

0.00E+00

-6.22E-01

3.22E-02

-5.89E-01

1.08E-02

0.00E+00

-5.40E-01

3.47E-02

-4.32E-01

1.62E-01

0.00E+00

-6.52E-01

3.16E-01

-4.22E-01

1.93E-01

0.00E+00

Appendix

TABLE X: Final constraint-violation comparison on the 15 CEC2025 CMOP functions. For each algorithm, the mean and SD over 30 runs are reported;

W

gives the Wilcoxon outcome of RDEx-CMOP against the competitor.

Multi-Objective Evolutionary Optimization

Multi-Objective Evolutionary Optimization uses population-based search to find diverse Pareto-optimal solutions when objectives conflict.

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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

Mar 28, 2026J/K move · Enter open · S save

RDEx-MOP: Indicator-Guided Reconstructed Differential Evolution for Fixed-Budget Multiobjective Optimization

Sichen Tao, Yifei Yang, Ruihan Zhao, Kaiyu Wang+2

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

Multi-Objective Evolutionary Optimization Evolutionary Optimization Differential Evolution

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Multiobjective optimisation in the CEC 2025 MOP track is evaluated not only by final IGD values but also by how quickly an algorithm reaches the target region under a fixed evaluation budget. This report documents RDEx-MOP, the reconstructed differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session) bound-constrained multiobjective track. RDEx-MOP integrates indicator-based environmental selection, a niche-maintained Pareto-candidate set, and complementary differential evolution operators for exploration and exploitation. We evaluate RDEx-MOP on the official CEC 2025 MOP benchmark using the released checkpoint traces and the median-target U-score framework. Experimental results show that RDEx-MOP achieves the highest total score and the best average rank among all released comparison algorithms, including the earlier RDEx baseline.

Abstract

Multi-Objective Evolutionary Optimization

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