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

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

cs.NEMar 28, 2026

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

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.

Figures & tables

Rank	Algorithm	Total Score	Avg Score/Prob.	Speed	Accuracy
1	RDEx-MOP	36343.5	3634.35	36343.5	0.0
2	RDEx	35956.5	3595.65	35956.5	0.0
3	TFBCEIBEA	15439.0	1543.90	7196.5	8242.5
4	TEMOFNSGA3	13811.5	1381.15	3811.5	10000.0
5	TGFMMOEA	11690.5	1169.05	2633.0	9057.5

TABLE I: CEC 2025 MOP evaluation (median target): overall scores over 10 problems and 30 runs for all released comparison algorithms.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Rank	Algorithm	Total Rank	Avg Rank/Prob.	Avg Speed	Avg Accuracy
1	RDEx-MOP	14.0	1.40	1.40	4.50
2	RDEx	16.0	1.60	1.60	4.50
3	TFBCEIBEA	34.0	3.40	3.50	2.50
4	TEMOFNSGA3	42.0	4.20	4.15	1.50
5	TGFMMOEA	44.0	4.40	4.35	2.00

Appendix

TABLE II: CEC 2025 MOP evaluation (median target): average rankings over 10 problems (lower is better) for all released comparison algorithms.

Competitor	Final IGD			TTT			AUC
Competitor	W/T/L	Holm	$A_{12}$	W/T/L	Holm	$A_{12}$	W/T/L	Holm	$A_{12}$
RDEx	2/8/0	0/10/0	0.55	2/8/0	0/10/0	0.52	0/9/1	0/10/0	0.45
TEMOFNSGA3	10/0/0	10/0/0	1.00	10/0/0	10/0/0	1.00	10/0/0	10/0/0	1.00
TFBCEIBEA	10/0/0	10/0/0	1.00	10/0/0	10/0/0	1.00	10/0/0	10/0/0	1.00
TGFMMOEA	10/0/0	10/0/0	1.00	10/0/0	10/0/0	1.00	10/0/0	10/0/0	1.00

Appendix

TABLE III: Complementary pairwise summary over the 10 CEC2025 MOP functions (30 runs). For each metric (Final IGD, TTT, and AUC), 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	TTT	AUC
RDEx-MOP	1.40	1.50	1.80
RDEx	1.60	1.50	1.20
TEMOFNSGA3	4.00	4.00	4.20
TFBCEIBEA	3.80	4.00	3.70
TGFMMOEA	4.20	4.00	4.10

Appendix

TABLE IV: Complementary Friedman tests on per-function medians over the 10 CEC2025 MOP functions (30 runs). Final IGD:

\chi^{2}=30.40

df=4

p=7.30E-06

; TTT:

\chi^{2}=30.00

df=4

p=8.63E-06

; AUC:

\chi^{2}=31.28

df=4

p=5.07E-06

. Lower average rank indicates better performance for each metric.

Problem

RDEx-MOP

RDEx

TEMOFNSGA3

TFBCEIBEA

TGFMMOEA

Mean

1.06E+01

1.34E-02

1.06E+01

1.37E-02

1.51E+01

1.42E+00

1.34E+01

9.01E-01

1.54E+01

1.70E+00

4.22E-02

1.95E-03

4.30E-02

2.47E-03

2.42E+00

1.56E+00

3.10E+00

3.15E+00

5.13E+01

3.09E+01

4.29E-01

3.12E-01

3.66E-01

2.81E-01

1.67E+01

1.13E+00

1.75E+01

9.05E-01

1.77E+01

1.15E+00

4.09E-01

3.44E-05

4.09E-01

3.48E-05

4.56E-01

1.32E-01

8.40E-01

4.97E-01

4.46E-01

9.71E-02

1.09E-01

3.17E-02

1.25E-01

3.84E-02

1.13E+00

7.09E-01

1.07E+00

6.98E-01

1.60E+00

7.10E-01

Appendix

TABLE V: Final IGD comparison on the 10 CEC2025 MOP functions. For each algorithm, the mean and SD over 30 runs are reported;

W

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

Problem	RDEx-MOP		RDEx			TEMOFNSGA3			TFBCEIBEA			TGFMMOEA
Problem	Mean	SD	Mean	SD	W	Mean	SD	W	Mean	SD	W	Mean	SD	W
1	4.4	0.9	4.3	1.3	=	490.9	42.0	+	406.2	149.1	+	489.5	44.1	+
2	5.0	1.1	5.1	1.1	=	457.1	98.1	+	405.7	172.2	+	472.0	108.9	+
3	2.3	0.4	2.6	0.7	+	411.5	137.0	+	474.7	100.6	+	461.2	101.1	+
4	2.5	0.5	2.7	0.5	=	469.3	62.1	+	483.0	59.2	+	474.8	70.1	+
5	3.5	0.5	3.3	0.5	=	454.2	97.5	+	399.6	161.8	+	490.4	43.6	+

Appendix

TABLE VI: Time-to-target comparison on the 10 CEC2025 MOP functions. TTT is the first checkpoint index (1–500) where the run reaches the median target (smaller is better); runs that never reach the target are assigned 501.

Problem	RDEx-MOP		RDEx			TEMOFNSGA3			TFBCEIBEA			TGFMMOEA
Problem	Mean	SD	Mean	SD	W	Mean	SD	W	Mean	SD	W	Mean	SD	W
1	0.00	0.00	0.00	0.00	=	0.60	0.17	+	0.31	0.17	+	0.61	0.16	+
2	0.00	0.00	0.00	0.00	=	0.51	0.23	+	0.45	0.34	+	1.37	0.31	+
3	0.00	0.00	0.00	0.00	=	0.34	0.14	+	0.40	0.16	+	0.48	0.13	+
4	0.00	0.00	0.00	0.00	=	0.09	0.05	+	0.16	0.09	+	0.08	0.06	+
5	0.00	0.00	0.00	0.00	=	0.28	0.11	+	0.25	0.13	+	0.39	0.08	+

Appendix

TABLE VII: Anytime convergence comparison using AUC over 500 checkpoints on the 10 CEC2025 MOP functions. For each run, AUC is computed as the mean of

\log_{10}(1+\max(f_{t}-\mathrm{TGT},0))

across checkpoints (smaller is better).

Multi-Objective Evolutionary Optimization

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

See all 85 papers

RDEx-CMOP: Feasibility-Aware Indicator-Guided Differential Evolution for Fixed-Budget Constrained Multiobjective Optimization
Apr 4, 2026Sichen Tao, Yifei Yang, Ruihan Zhao +3Multi-Objective Evolutionary Optimization Multi-Objective Optimization
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
RDEx-SOP: Exploitation-Biased Reconstructed Differential Evolution for Fixed-Budget Bound-Constrained Single-Objective Optimization
Mar 28, 2026Sichen Tao, Yifei Yang, Ruihan Zhao +3Differential Evolution Constrained Optimization
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
RDEx-CSOP: Feasibility-Aware Reconstructed Differential Evolution with Adaptive epsilon-Constraint Ranking
Mar 28, 2026Sichen Tao, Yifei Yang, Ruihan Zhao +3Constrained Optimization Differential Evolution
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

Apr 4, 2026J/K move · Enter open · S save

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

Open paper AskPDF arXiv

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.

Abstract

Multi-Objective Evolutionary Optimization

Explore similar work

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

RDEx-SOP: Exploitation-Biased Reconstructed Differential Evolution for Fixed-Budget Bound-Constrained Single-Objective Optimization

RDEx-CSOP: Feasibility-Aware Reconstructed Differential Evolution with Adaptive epsilon-Constraint Ranking