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RDEx-MOP: Indicator-Guided Reconstructed Differential Evolution for Fixed-Budget Multiobjective Optimization
Sichen Tao, Yifei Yang, Ruihan Zhao, Kaiyu Wang, Sicheng Liu, Shangce Gao
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Summary
RDEx-MOP is a reconstructed differential evolution algorithm designed for fixed-budget multiobjective optimization, specifically for the CEC 2025 competition. It utilizes indicator-based environmental selection, a niche-maintained Pareto-candidate set, and specialized DE operators (current-to-ppbest/1 and niche exploration) to achieve superior convergence and diversity, ranking first in the official U-score evaluation.
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RDEx-MOP → achievedrank → 1
confidence 100% · RDEx-MOP achieves the highest total score and the best average rank among all released comparison algorithms
RDEx-MOP → competedin → CEC 2025
confidence 100% · This report documents RDEx-MOP, the reconstructed differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition
RDEx-MOP → optimizedfor → IGD
confidence 95% · Multiobjective optimisation in the CEC 2025 MOP track is evaluated not only by final IGD values
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Abstract
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.
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- Source: https://arxiv.org/abs/2603.27092v1
- Canonical: https://arxiv.org/abs/2603.27092v1
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RDEx-MOP: Indicator-Guided Reconstructed Differential Evolution for Fixed-Budget Multiobjective Optimization Sichen Tao1,2, Yifei Yang3, Ruihan Zhao4,5, Kaiyu Wang6,7, Sicheng Liu8, Shangce Gao1 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. Index Terms: Multiobjective Optimisation, Differential Evolution, CEC 2025, IGD, U-score I Introduction Multiobjective optimisation requires a solver to approximate the Pareto front with both good convergence and adequate diversity. NSGA-I, SPEA2, IBEA, and MOEA/D established four influential paradigms—non-dominated sorting, strength-based fitness, indicator-based comparison, and decomposition—that still shape modern evolutionary multiobjective algorithms [4, 15, 14, 13]. Their common lesson is that archive quality depends not only on how solutions are compared in the objective space, but also on how diversity pressure is maintained during the run. For continuous decision spaces, differential evolution (DE) is an attractive search engine because it provides directional variation at relatively low algorithmic complexity [8]. In parallel with the classical EMO selectors, a recent Zhengzhou-University-centred line has pushed DE further into multimodal and fixed-budget MOP search through archive-assisted maintenance, decision-space diversity control, and two-stage population management [1, 6, 5, 2]. In particular, TEMOF showed that a two-stage framework can substantially improve fixed-budget IGD behaviour on the 2024 competition suite [2]. These studies are directly relevant here because they bridge environmental selection in the objective space with directional search pressure in the decision space. At the same time, the single-objective DE line from JADE to SHADE and L-SHADE established that top-ranked current-to-ppbest variation, shrinking elite windows, and moderate heavy-tailed perturbation can improve fixed-budget search without collapsing diversity [12, 11, 10, 3]. The key point for MOPs is not to transplant those mechanisms mechanically, but to retain only the parts that strengthen early IGD descent while leaving archive maintenance to the multiobjective selector. The CEC 2025 MOP track makes this balance explicit because performance is judged not only by the final IGD value, but also by how rapidly the algorithm reaches target regions along the run under the U-score framework [7]. Hence a competitive algorithm must do more than deliver a good final front: it must reduce IGD quickly, keep sparse regions alive, and remain stable under the released official evaluation package [9]. RDEx-MOP follows this logic by reconstructing the 2024 TEMOF-style backbone with a stronger top-ranked DE search pressure and lightweight Cauchy refinement. It combines indicator-based environmental selection, a niche-maintained Pareto-candidate set, a current-to-ppbest/1 exploitation operator, and a complementary niche exploration operator. The main contribution is a reconstructed DE framework targeted at fixed-budget multiobjective benchmarking, where rapid IGD reduction and stable diversity must be achieved simultaneously. The source code for RDEx-MOP is publicly available on Sichen Tao’s GitHub page: https://github.com/SichenTao. I Benchmark and Evaluation (CEC 2025 MOP) The CEC 2025 bound-constrained multiobjective track contains 10 benchmark problems (MaOP1–MaOP10) released in the official competition package [9]. Each problem is evaluated with 30 independent runs. The maximum evaluation budget is MaxFE=100000MaxFE=100000, and the platform records progress at 500 checkpoints per run (every 200 function evaluations). I-A Multiobjective Formulation The general MOP can be written as: minx∈ℝDF(x)=(f1(x),f2(x),…,fM(x)), _x ^DF(x)=(f_1(x),f_2(x),…,f_M(x)), (1) subject to bound constraints ℓj≤xj≤uj _j≤ x_j≤ u_j for j=1,…,Dj=1,…,D. I-B IGD Indicator Let P be the set of solutions returned by an algorithm and P⋆P be a reference set that approximates the true Pareto front. IGD is defined as: IGD(P,P⋆)=1|P⋆|∑y∈P⋆minx∈P∥F(x)−y∥2.IGD(P,P )= 1 P _y∈ P _x∈ P F(x)-y _2. (2) Smaller IGD values indicate better convergence and diversity with respect to the reference set. I RDEx-MOP Algorithm RDEx-MOP is implemented in a PlatEMO-style framework and reconstructs the recent two-stage MOP competition backbone [2] with a stronger top-ranked DE search operator. It maintains three main populations during the run: (i) a working population P, (i) an auxiliary population P2P_2 (used after half of the budget with a random switching rule), and (i) a Pareto-candidate set PCPC obtained from the current nondominated solutions with a niche-based maintenance procedure related to recent archive- and niche-oriented MMOP designs [1, 6, 5]. The algorithm alternates between a niche exploration operator and an exploitation operator based on a DE/current-to-ppbest/1 mutation with lightweight Cauchy perturbations [8]. I-A Indicator-based Fitness Assignment RDEx-MOP adopts an indicator-based fitness assignment as in IBEA [14]. Given a population of size N with objective vectors F(xi)i=1N\F(x_i)\_i=1^N, objectives are first normalised component-wise to [0,1][0,1]. Then an additive indicator matrix I∈ℝN×NI ^N× N is computed: I(i,j)=maxm∈1,…,M(fm(xi)−fm(xj)),I(i,j)= _m∈\1,…,M\ (f_m(x_i)-f_m(x_j) ), (3) and the scaling vector C is defined by C(j)=maxi|I(i,j)|C(j)= _i I(i,j) . The fitness of solution xix_i is: fit(xi)=∑j=1N(−exp(−I(i,j)C(j)κ))+1,fit(x_i)= _j=1^N (- (- I(i,j)C(j)κ ) )+1, (4) where κ is the indicator parameter (default κ=0.05κ=0.05). I-B Indicator-based Environmental Selection Given a candidate set S, RDEx-MOP repeatedly removes the worst individual according to Eq. (4) until ||=N =N. After deleting xwx_w, the fitness values of the remaining individuals are updated by fit(xi)←fit(xi)+exp(−I(w,i)C(w)κ),xi∈∖xw.fit(x_i) (x_i)+ \! (- I(w,i)C(w)κ ), x_i \x_w\. (5) This selection provides stable convergence pressure without requiring explicit crowding-distance sorting. I-C Pareto-candidate Set and Niche Maintenance The Pareto-candidate set PCPC is formed by extracting the first nondominated front from a pooled set using nondominated sorting. If |PC|>N PC >N, a niche maintenance procedure deletes solutions one by one to promote diversity in the objective space. After normalisation, the niche radius is estimated by the mean distance to the 33rd nearest neighbour, r0=1|PC|∑idi(3),r_0= 1 PC _id_i^(3), (6) and the pairwise crowding matrix is set to Rij=min(dij/r0,1)R_ij= (d_ij/r_0,1). The deletion step removes the solution with the largest multiplicative crowding score 1−∏jRij1- _jR_ij. I-D Niche Exploration Operator Let nND=|PC|n_ND= PC . RDEx-MOP detects solutions in PCPC that reside in sparse niches relative to the current working population P, following the same design intention as recent niche-exploration MMODE variants [6, 5]. After normalisation, a niche threshold is set to r=nNDNr0,r0=mean(di(3)),r= n_NDNr_0, r_0=mean (d^(3)_i ), (7) where di(3)d^(3)_i is the distance from solution i to its 33rd nearest neighbour in PCPC. Solutions that have at most one neighbour within radius r are selected for exploration. For each selected solution, a second parent is sampled uniformly from the current population, and a lightweight “half-DE” operator generates an offspring via v=x+F⋅(xr1−xg),v=x+F·(x_r_1-x_g), (8) followed by binomial crossover with CR∼(0.5,0.1)CR (0.5,0.1) clipped to [0,1][0,1], where F∼Cauchy(0.7,0.2)∩[0,1]F (0.7,0.2)∩[0,1] and xgx_g is taken from the paired parent set. I-E Exploitation Operator (DE/current-to-ppbest/1 + Cauchy Perturbation) RDEx-MOP uses a discrete parameter pool: F∈0.6,0.8,1.0,CR∈0.1,0.2,1.0.F∈\0.6,0.8,1.0\, CR∈\0.1,0.2,1.0\. (9) The p-best selection window shrinks during the run: p=max(2,⌊0.17N(1−0.9FEMaxFE)+0.5⌋),p= \! (2, 0.17N (1-0.9 FEMaxFE )+0.5 ), (10) and the donor vector is generated by: v=x+F⋅(xpbest−x)+F⋅(xr1−xr2).v=x+F·(x_pbest-x)+F·(x_r_1-x_r_2). (11) After crossover, each decision component is perturbed with probability 0.20.2 using Cauchy(uj,0.2)Cauchy(u_j,0.2), and then clipped to the bounds. I-F Overall Procedure Algorithm 1 sketches the PlatEMO implementation used in this repository. Input: Population size N, indicator parameter κ, evaluation budget MaxFEMaxFE. Output: Final Pareto-candidate set PCPC. 1 P←P← random initial population of size N; 2 [PC,nND]←Selection(P,N)[PC,n_ND]← Selection(P,N); 3 P2←PCP_2← PC; 4 while FE<MaxFEFE<MaxFE do 5 Q←Exploration(PC,P,nND,N)Q← Exploration(PC,P,n_ND,N); 6 if FE≥0.5MaxFEFE≥ 0.5MaxFE and rand<0.5rand<0.5 then 7 P2←EnvSel(P2∪Q,N,κ)P_2← EnvSel(P_2∪ Q,N,κ); 8 R←OperatorDE(P2,κ)R← OperatorDE(P_2,κ); 9 P2←EnvSel(P2∪R,N,κ)P_2← EnvSel(P_2∪ R,N,κ); 10 11 else 12 P←EnvSel(P∪Q,N,κ)P← EnvSel(P∪ Q,N,κ); 13 R←OperatorDE(P,κ)R← OperatorDE(P,κ); 14 P←EnvSel(P∪R,N,κ)P← EnvSel(P∪ R,N,κ); 15 16 [PC,nND]←Selection(PC∪R∪Q∪P2,N)[PC,n_ND]← Selection(PC∪ R∪ Q∪ P_2,N); 17 P2←PCP_2← PC; 18 Algorithm 1 RDEx-MOP (high level). I-G Computational Complexity The indicator-based fitness assignment requires O(N2M)O(N^2M) time to compute the pairwise indicator matrix and update fitness values. The niche-based maintenance and exploration operators rely on pairwise distances in the objective space and are also O(N2M)O(N^2M) per generation. IV Experimental Results IV-A Protocol For each problem, we execute 30 independent runs under MaxFE=100000MaxFE=100000 with population size N=100N=100. The platform records IGD every 200 evaluations, yielding 500 checkpoints per run. IV-B Parameter Settings Unless otherwise stated, RDEx-MOP uses the reference configuration embedded in the released competition code: population size N=100N=100, indicator parameter κ=0.05κ=0.05, the discrete F/CRF/CR pool in Eq. (9), and Cauchy perturbation probability 0.20.2. IV-C Experimental Settings RDEx-MOP is evaluated with the official median-target U-score framework. We compare RDEx-MOP with all remaining algorithms available in the released competition package [9]: RDEx, TEMOFNSGA3, TFBCEIBEA, and TGFMMOEA. In the released evaluation files, the submitted winner is labelled as MORDEx; for naming consistency, we report it as RDEx-MOP throughout this manuscript. IV-D Statistical Results IV-D1 Overall U-score Results Table I reports the official median-target U-score results for all released comparison algorithms. TABLE I: CEC 2025 MOP evaluation (median target): overall scores over 10 problems and 30 runs for all released comparison algorithms. 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 RDEx-MOP achieves the highest total score (36343.536343.5) and the best average rank (1.401.40). It remains ahead of the earlier RDEx baseline (35956.535956.5, average rank 1.601.60), while all other competitors are far behind in the official U-score totals. Both RDEx variants reach the median target on all runs, so their U-scores are dominated by the Speed category and the residual Accuracy scores become zero. IV-D2 Supplementary Diagnostics To keep the main paper focused on the official competition metric, all per-function final-IGD and speed-side diagnostics are moved to the appendix. Appendix A first reports the official all-algorithm U-score ranking table and then collects complementary pairwise and Friedman analyses for final IGD, T, and AUC. V Conclusion RDEx-MOP is an indicator-guided differential evolution framework for the CEC 2025 bound-constrained multiobjective track. By combining indicator-based environmental selection, a niche-maintained Pareto-candidate set, and complementary DE operators for exploration and exploitation, the method achieves first-place official U-score performance on the full released benchmark suite. Acknowledgment This research was partially supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI under Grant JP22H03643, Japan Science and Technology Agency (JST) Support for Pioneering Research Initiated by the Next Generation (SPRING) under Grant JPMJSP2145, and JST through the Establishment of University Fellowships towards the Creation of Science Technology Innovation under Grant JPMJFS2115. References [1] P. Chen, Z. Li, K. Qiao, P. N. Suganthan, X. Ban, K. Yu, C. Yue, and J. Liang (2024) An archive-assisted multi-modal multi-objective evolutionary algorithm. Swarm and Evolutionary Computation 89, p. 101738. External Links: Document Cited by: §I, §I. [2] P. Chen, J. Liang, K. Qiao, P. N. Suganthan, and X. Ban (2024) A two-stage evolutionary framework for multi-objective optimization. In 2024 IEEE Congress on Evolutionary Computation (CEC), p. 1–8. External Links: Document Cited by: §I, §I. [3] T. J. Choi and C. W. Ahn (2021) An improved LSHADE-RSP algorithm with the Cauchy perturbation: iLSHADE-RSP. Knowledge-Based Systems 215, p. 106628. Cited by: §I. [4] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan (2002) A fast and elitist multiobjective genetic algorithm: NSGA-I. IEEE Transactions on Evolutionary Computation 6 (2), p. 182–197. Cited by: §I. [5] J. Liang, X. Sui, C. Yue, M. Yu, G. Li, and M. Li (2024) Multimodal multiobjective differential evolution algorithm based on enhanced decision space search. Swarm and Evolutionary Computation 86, p. 101682. External Links: Document Cited by: §I, §I-D, §I. [6] H. Lin, J. Liang, C. Yue, and Y. Wang (2024) A niching-based reproduction and preselection-based multiobjective differential evolution for multimodal multiobjective optimization. In 2024 IEEE Congress on Evolutionary Computation (CEC), p. 1–8. External Links: Document Cited by: §I, §I-D, §I. [7] K. V. Price, A. Kumar, and P. N. Suganthan (2023) Trial-based dominance for comparing both the speed and accuracy of stochastic optimizers with standard non-parametric tests. Swarm and Evolutionary Computation 78, p. 101287. Cited by: §I. [8] R. Storn and K. Price (1997) Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11, p. 341–359. Cited by: §I, §I. [9] P. N. Suganthan (2025) 2025 CEC Competition Repository. Note: GitHub repositoryAccessed: 2026-03-09 External Links: Link Cited by: §I, §I, §IV-C. [10] R. Tanabe and A. S. Fukunaga (2014) Improving the search performance of SHADE using linear population size reduction. In 2014 IEEE Congress on Evolutionary Computation (CEC), p. 1658–1665. Cited by: §I. [11] R. Tanabe and A. Fukunaga (2013) Success-history based parameter adaptation for differential evolution. In 2013 IEEE Congress on Evolutionary Computation (CEC), p. 71–78. Cited by: §I. [12] J. Zhang and A. C. Sanderson (2009) JADE: adaptive differential evolution with optional external archive. IEEE Transactions on Evolutionary Computation 13 (5), p. 945–958. Cited by: §I. [13] Q. Zhang and H. Li (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation 11 (6), p. 712–731. Cited by: §I. [14] E. Zitzler and S. Künzli (2004) Indicator-based selection in multiobjective search. In Parallel Problem Solving from Nature – PPSN VIII, p. 832–842. Cited by: §I, §I-A. [15] E. Zitzler, M. Laumanns, and L. Thiele (2001) SPEA2: improving the strength Pareto evolutionary algorithm. In Evolutionary Methods for Design, Optimisation and Control with Applications to Industrial Problems (EUROGEN 2001), p. 95–100. Cited by: §I. Appendix A Supplementary U-score Tables TABLE I: CEC 2025 MOP evaluation (median target): average rankings over 10 problems (lower is better) for all released comparison algorithms. 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 B Complementary Diagnostics The tables in this section are diagnostic supplements to the official U-score results and help explain final-value and speed-side behaviour. TABLE I: Complementary pairwise summary over the 10 CEC2025 MOP functions (30 runs). For each metric (Final IGD, T, and AUC), we report uncorrected per-function Wilcoxon W/T/L at α=0.05α=0.05, Holm-corrected W/T/L across functions, and the median Vargha–Delaney A12A_12 effect size (larger is better for minimization). Competitor Final IGD T AUC W/T/L Holm A12A_12 W/T/L Holm A12A_12 W/T/L Holm A12A_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 TABLE IV: Complementary Friedman tests on per-function medians over the 10 CEC2025 MOP functions (30 runs). Final IGD: χ2=30.40χ^2=30.40, df=4df=4, p=7.30E−06p=7.30E-06; T: χ2=30.00χ^2=30.00, df=4df=4, p=8.63E−06p=8.63E-06; AUC: χ2=31.28χ^2=31.28, df=4df=4, p=5.07E−06p=5.07E-06. Lower average rank indicates better performance for each metric. Algorithm Final IGD T 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 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 Mean SD Mean SD W Mean SD W Mean SD W Mean SD W 1 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 + 2 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 + 3 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 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 + 5 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 + 6 1.09E-01 2.93E-03 1.10E-01 3.10E-03 = 1.19E+00 3.41E-01 + 7.62E-01 1.95E-01 + 2.13E+00 1.96E+00 + 7 6.82E-02 2.54E-03 6.94E-02 2.24E-03 = 7.91E+00 6.99E+00 + 6.50E+00 6.31E+00 + 5.94E+00 6.07E+00 + 8 7.54E-02 2.86E-03 7.76E-02 3.53E-03 + 7.69E+00 6.50E+00 + 4.73E+00 6.11E+00 + 3.68E+00 3.28E+00 + 9 1.01E-01 3.58E-03 1.02E-01 3.31E-03 = 7.72E+00 4.84E+00 + 8.72E+00 8.22E+00 + 5.28E+00 4.19E+00 + 10 9.97E-02 3.56E-03 1.02E-01 3.16E-03 + 7.27E+00 5.50E+00 + 7.54E+00 7.08E+00 + 5.29E+00 4.04E+00 + W/T/L −/−/−-/-/- 2/8/0 10/0/0 10/0/0 10/0/0 TABLE VI: Time-to-target comparison on the 10 CEC2025 MOP functions. T 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 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 + 6 3.9 0.7 3.9 0.7 = 493.3 40.0 + 407.7 123.3 + 493.6 40.0 + 7 8.1 1.4 7.8 1.4 = 482.7 81.4 + 454.2 66.6 + 484.2 44.3 + 8 8.5 1.7 9.1 2.1 = 483.0 68.3 + 424.2 104.0 + 480.1 64.0 + 9 5.5 1.2 6.3 1.4 + 464.9 104.7 + 404.1 146.6 + 468.2 98.6 + 10 4.9 1.1 4.5 1.1 = 476.2 91.1 + 462.2 105.9 + 477.6 55.3 + W/T/L −/−/−-/-/- 2/8/0 10/0/0 10/0/0 10/0/0 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 log10(1+max(ft−TGT,0)) _10(1+ (f_t-TGT,0)) across checkpoints (smaller is better). Problem RDEx-MOP RDEx TEMOFNSGA3 TFBCEIBEA TGFMMOEA 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 + 6 0.00 0.00 0.00 0.00 = 0.25 0.08 + 0.14 0.06 + 0.43 0.12 + 7 0.01 0.00 0.01 0.00 - 0.93 0.33 + 0.90 0.29 + 0.87 0.27 + 8 0.01 0.00 0.01 0.00 = 0.91 0.41 + 0.76 0.29 + 0.77 0.28 + 9 0.01 0.00 0.01 0.00 = 0.88 0.39 + 0.86 0.39 + 0.76 0.28 + 10 0.01 0.00 0.01 0.00 = 0.87 0.35 + 0.78 0.37 + 0.77 0.28 + W/T/L −/−/−-/-/- 0/9/1 10/0/0 10/0/0 10/0/0