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RDEx-CSOP: Feasibility-Aware Reconstructed Differential Evolution with Adaptive epsilon-Constraint Ranking
Sichen Tao, Yifei Yang, Ruihan Zhao, Kaiyu Wang, Sicheng Liu, Shangce Gao
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Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 95%
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Summary
RDEx-CSOP is a feasibility-aware variant of Differential Evolution designed for constrained single-objective numerical optimization. It integrates success-history parameter adaptation, an exploitation-biased hybrid search, and a time-varying epsilon-constraint handling mechanism. Evaluated on the CEC 2025 CSOP benchmark, it achieved the highest total U-score and best average rank among comparison algorithms, demonstrating superior speed and competitive constraint-handling performance.
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RDEx-CSOP → evaluatedon → CEC 2025 CSOP
confidence 100% · We evaluate RDEx-CSOP on the official CEC 2025 CSOP benchmark
RDEx-CSOP → implements → Differential Evolution
confidence 95% · RDEx-CSOP, a constrained differential evolution variant
Sichen Tao → developed → RDEx-CSOP
confidence 90% · The source code for RDEx-CSOP is publicly available on Sichen Tao’s GitHub page
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Abstract
Abstract:Constrained single-objective numerical optimisation requires both feasibility maintenance and strong objective-value convergence under limited evaluation budgets. This report documents RDEx-CSOP, a constrained differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session). RDEx-CSOP combines success-history parameter adaptation with an exploitation-biased hybrid search and an {\epsilon}-constraint handling mechanism with a time-varying threshold. We evaluate RDEx-CSOP on the official CEC 2025 CSOP benchmark using the U-score framework (Speed, Accuracy, and Constraint categories). The results show that RDEx-CSOP achieves the highest total score and the best average rank among all released comparison algorithms, mainly through strong speed and competitive constraint-handling performance across the 28 benchmark functions.
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- Source: https://arxiv.org/abs/2603.27090v1
- Canonical: https://arxiv.org/abs/2603.27090v1
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RDEx-CSOP: Feasibility-Aware Reconstructed Differential Evolution with Adaptive ε -Constraint Ranking Sichen Tao1,2, Yifei Yang3, Ruihan Zhao4,5, Kaiyu Wang6,7, Sicheng Liu8, Shangce Gao1 Abstract Constrained single-objective numerical optimisation requires both feasibility maintenance and strong objective-value convergence under limited evaluation budgets. This report documents RDEx-CSOP, a constrained differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session). RDEx-CSOP combines success-history parameter adaptation with an exploitation-biased hybrid search and an ε -constraint handling mechanism with a time-varying threshold. We evaluate RDEx-CSOP on the official CEC 2025 CSOP benchmark using the U-score framework (Speed, Accuracy, and Constraint categories). The results show that RDEx-CSOP achieves the highest total score and the best average rank among all released comparison algorithms, mainly through strong speed and competitive constraint-handling performance across the 28 benchmark functions. Index Terms: Differential Evolution, Constrained Optimisation, CEC 2025, Single-objective, ε -constraint, U-score I Introduction Differential evolution (DE) and its adaptive descendants are among the most competitive optimisers for continuous real-parameter search because they combine strong directional variation with a lightweight algorithmic structure [9, 14, 12, 11]. Reviewing studies on parameter control further show that the SHADE/L-SHADE line is one of the most reliable fixed-budget design families in modern DE [13]. In unconstrained single-objective optimisation, this line has progressively improved search quality by strengthening parameter adaptation, selective pressure, and local perturbation, leading to variants such as LSHADE-RSP, iLSHADE-RSP, EB-LSHADE, and LSRTDE [6, 1, 4, 7, 8]. These studies collectively show that both operator design and parameter learning are decisive when the evaluation budget is tight. Constrained single-objective optimisation adds a second layer of difficulty because the algorithm must not only improve objective values, but also reach and preserve feasibility efficiently. The constrained-optimisation literature has repeatedly shown that feasibility rules and ε -level control are not superficial add-ons: they redefine how candidate solutions are ranked, which solutions survive, and therefore what learning signal is available to an adaptive optimiser [2, 10, 3]. As a result, a strong bound-constrained DE cannot simply be transplanted unchanged into constrained search; mutation bias, replacement, and parameter memories must all remain compatible with feasibility-aware ranking. This issue becomes sharper in the CEC 2025 CSOP track, where algorithms are evaluated under the U-score framework rather than by final objective values alone [5]. In this regime, a method that is accurate but slow to attain feasibility, or fast but unstable after becoming feasible, can both be penalised. What is needed is therefore not only a strong DE operator set, but also a constraint-handling mechanism that shapes the search trajectory early enough to improve both feasible target-attainment speed and final solution quality. RDEx-CSOP is designed from this perspective as a feasibility-aware reconstruction of the recent competition DE line. It integrates adaptive ε -ranking directly into selection and learning, while combining success-history memories, an exploitation-biased hybrid mutation branch, and success-rate-guided scaling-factor sampling. The main contribution is therefore a compact constrained DE framework in which operator adaptation is explicitly aligned with feasibility-aware search dynamics, with the goal of accelerating feasible progress without sacrificing final objective quality. The source code for RDEx-CSOP is publicly available on Sichen Tao’s GitHub page: https://github.com/SichenTao. I Reconstructed Differential Evolution with ε -constraint Handling (RDEx-CSOP) I-A Problem Formulation RDEx-CSOP solves a constrained single-objective optimisation problem: minx∈ℝD _x ^D f(x), f(x), (1) s.t. gi(x)≤0,i=1,…,mg, g_i(x)≤ 0, i=1,…,m_g, (2) hj(x)=0,j=1,…,mh, h_j(x)=0, j=1,…,m_h, (3) ℓk≤xk≤uk,k=1,…,D. _k≤ x_k≤ u_k, k=1,…,D. (4) Following the released implementation, RDEx-CSOP uses the averaged violation measure ϕ(x)=1mg+mh(∑i=1mgmax0,gi(x)+∑j=1mhmax0,|hj(x)|−εeq),φ(x)= 1m_g+m_h aligned (& _i=1^m_g \0,g_i(x)\\\ &+ _j=1^m_h \0, h_j(x) - _eq\ ), aligned (5) where εeq=10−4 _eq=10^-4 is the equality tolerance. I-B Adaptive ε -ranking and Selection Let ℱ(g)=xi(g)i=1N(g)F^(g)=\x_i^(g)\_i=1^N^(g) denote the current elite front. RDEx-CSOP sets the threshold from the empirical penalty distribution of ℱ(g)F^(g), following the standard rationale of ε -level constrained search [10]: k(g)=max(1,⌊ηN(g)(1−NFEMaxFE)2⌋),η=0.8,k^(g)= \! (1, η N^(g) (1- NFEMaxFE )^2 ), η=0.8, (6) ε(g)=ϕ(k(g))(g),NFE≤0.8MaxFE,0,otherwise, ^(g)= cases _(k^(g))^(g),&NFE≤ 0.8\,MaxFE,\\ 0,&otherwise, cases (7) where ϕ(k)(g) _(k)^(g) is the k-th order statistic of ϕ(xi(g))i=1N(g)\φ(x_i^(g))\_i=1^N^(g). The corresponding ranking score is s(g)(x)=f(x),ϕ(x)≤ε(g),fmax(g)+1+ϕ(x),otherwise,s^(g)(x)= casesf(x),&φ(x)≤ ^(g),\\ f_ ^(g)+1+φ(x),&otherwise, cases (8) with fmax(g)=maxx∈ℱ(g)f(x)f_ ^(g)= _x ^(g)f(x). During one-to-one selection, let ϕ~(x)=0 φ(x)=0 if ϕ(x)≤ε(g)φ(x)≤ ^(g) and ϕ~(x)=ϕ(x) φ(x)=φ(x) otherwise. Then a trial ui(g)u_i^(g) replaces xi(g)x_i^(g) iff ϕ~(ui(g)) φ(u_i^(g)) <ϕ~(xi(g))or < φ(x_i^(g)) (9) ϕ~(ui(g)) φ(u_i^(g)) =ϕ~(xi(g))∧f(ui(g))≤f(xi(g)). = φ(x_i^(g)) f(u_i^(g))≤ f(x_i^(g)). I-C Standard Mutation and Crossover RDEx-CSOP maintains success-history memories MFM_F and MCRM_CR of size H. In the standard branch, Fi(g)∼(max(0,(SR(g))1/3),0.052)∩[0,1],F_i^(g) \! ( \! (0,(SR^(g))^1/3 ),0.05^2 )∩[0,1], (10) while CRi(g)∼(MCR,h,0.12)CR_i^(g) (M_CR,h,0.1^2) clipped to [0,1][0,1]. The success-rate-guided centre in Eq. (10) is motivated by the LSRTDE line [7, 8]. Using the ε -ranking in Eq. (8), RDEx-CSOP samples xpbest(g)x_pbest^(g) from the best p(g)=max(2,⌊0.3N(g)⌋)p^(g)= (2, 0.3N^(g) ) solutions and generates vi(g)=xi(g)+Fi(g)(xpbest(g)−xi(g))+Fi(g)(xr1(g)−xr2(g)),v_i^(g)=x_i^(g)+F_i^(g)\! (x_pbest^(g)-x_i^(g) )+F_i^(g)\! (x_r_1^(g)-x_r_2^(g) ), (11) where r1r_1 is sampled from the front with an exponential rank bias and r2r_2 from the current population. Binomial crossover is then applied: ui,j(g)=vi,j(g),randj<CRi(g)orj=jrand,x^i,j(g),otherwise,u_i,j^(g)= casesv_i,j^(g),&rand_j<CR_i^(g)\ or\ j=j_rand,\\ x_i,j^(g),&otherwise, cases (12) where x^i,j(g) x_i,j^(g) equals either xi,j(g)x_i,j^(g) or a local perturbation Cauchy(xi,j(g),0.1)Cauchy(x_i,j^(g),0.1) with probability 0.20.2, following the local-refinement idea used in iLSHADE-RSP [1]. If a trial component violates the bounds, midpoint repair is used: ui,j(g)←xi,j(g)+ℓj2,ui,j(g)<ℓj,xi,j(g)+uj2,ui,j(g)>uj.u_i,j^(g)← cases x_i,j^(g)+ _j2,&u_i,j^(g)< _j,\\ x_i,j^(g)+u_j2,&u_i,j^(g)>u_j. cases (13) I-D Exploitation-biased Branch and Memory Update The exploitation-biased (EB) branch is activated more often in the later search stage through an adaptive rate ρEB(g) _EB^(g), following the ordered-donor intuition of EB-LSHADE [4]. After ranking three sampled donors by Eq. (8) as xbest(g)x_best^(g), xmid(g)x_mid^(g), and xworst(g)x_worst^(g), the EB donor is vi(g)=xi(g)+Fi(g)(xbest(g)−xi(g))+Fi(g)(xmid(g)−xworst(g)).v_i^(g)=x_i^(g)+F_i^(g)\! (x_best^(g)-x_i^(g) )+F_i^(g)\! (x_mid^(g)-x_worst^(g) ). (14) In this branch, Fi(g)F_i^(g) is sampled from a truncated Cauchy distribution centred at MF,hM_F,h (or 0.40.4 for the fallback slot), and CRi(g)CR_i^(g) is sampled from (MCR,h,0.12)N(M_CR,h,0.1^2) (or from a fallback mean 0.90.9) with early-stage lower bounds CRi(g)≥0.7CR_i^(g)≥ 0.7 before 0.25MaxFE0.25\,MaxFE and CRi(g)≥0.6CR_i^(g)≥ 0.6 before 0.5MaxFE0.5\,MaxFE. Let Δi _i denote the improvement contributed by a successful trial, and define ΔEB(g)=∑i∈EB(g)Δi _EB^(g)= _i _EB^(g) _i and Δstd(g)=∑i∈std(g)Δi _std^(g)= _i _std^(g) _i. The hybrid rate is updated by ρEB(g+1)=ΔEB(g)ΔEB(g)+Δstd(g),if ΔEB(g),Δstd(g)>0,0.7,otherwise. _EB^(g+1)= cases _EB^(g) _EB^(g)+ _std^(g),&if _EB^(g), _std^(g)>0,\\[4.30554pt] 0.7,&otherwise. cases (15) If AiA_i is the realised crossover ratio of trial i and wi=Δi/∑tΔtw_i= _i/ _t _t, then successful parameters update the memories through MF M_F ←12(MF+∑iwiFi2∑iwiFi), ← 12\! (M_F+ _iw_iF_i^2 _iw_iF_i ), (16) MCR M_CR ←12(MCR+∑iwiAi2∑iwiAi). ← 12\! (M_CR+ _iw_iA_i^2 _iw_iA_i ). The elite front is reduced linearly from N0N_0 to Nmin=4N_ =4, again following the L-SHADE design principle [11]: N(g+1)=⌊N0+(Nmin−N0)NFEMaxFE⌋.N^(g+1)= N_0+(N_ -N_0) NFEMaxFE . (17) I-E Pseudocode Algorithm 1 sketches the overall RDEx-CSOP procedure. Input: N0N_0, memory size H, evaluation budget MaxFEMaxFE. Output: Best feasible solution found. 1 Initialise population/front, memories, and ρEB(0) _EB^(0); 2 while NFE<MaxFENFE<MaxFE do 3 Compute ε(g) ^(g) and ε -ranks by Eqs. (7)–(8); 4 foreach xix_i in the current front do 5 Sample (Fi,CRi)(F_i,CR_i) from the memories; 6 Generate viv_i by Eq. (11) or Eq. (14); 7 Apply crossover, perturbation, and repair by Eqs. (12)–(13); 8 Evaluate the trial and accept it by Eq. (9); 9 10 Update ρEB _EB and the memories by Eqs. (15) and (16); 11 Reduce the front by Eq. (17); 12 Algorithm 1 RDEx-CSOP framework. I Experimental Results I-A Benchmark Functions and Protocol The CEC 2025 CSOP track consists of 28 constrained problems with dimension D=30D=30. Each problem is evaluated with 25 independent runs and a maximum budget of MaxFE=20000×DMaxFE=20000× D function evaluations. The official U-score evaluation uses three categories: Speed, Accuracy, and Constraint, and aggregates results across problems under a fixed checkpoint schedule. I-B Parameter Settings Unless otherwise stated, RDEx-CSOP uses the released reference configuration: N0=600N_0=600, H=5H=5, initial memories MF=0.3M_F=0.3 and MCR=1.0M_CR=1.0, initial ρEB=0.7 _EB=0.7, η=0.8η=0.8 in Eq. (6), perturbation probability 0.20.2, and the linear front reduction in Eq. (17). I-C Experimental Settings RDEx-CSOP is evaluated with the official U-score framework using the median target setting. We compare RDEx-CSOP with all remaining algorithms available in the released competition package: RDEx, UDEIII, and CL-SRDE. In the released evaluation files, the submitted winner is labelled as CORDEx; for naming consistency, we report it as RDEx-CSOP throughout this manuscript. I-D Statistical Results I-D1 Overall U-score Results Table I reports the official median-target U-score results for all released comparison algorithms. TABLE I: CEC 2025 CSOP evaluation (median target): overall scores over 28 problems and 25 runs for all released comparison algorithms. Rank Algorithm Total Score Avg Score/Prob. Speed Accuracy Constraint 1 RDEx-CSOP 53680.5 1917.16 44437.0 3440.0 5803.5 2 RDEx 49155.0 1755.54 39695.5 3730.5 5729.0 3 UDEIII 30545.0 1090.89 22368.5 3093.0 5083.5 4 CL-SRDE 28805.5 1028.77 19673.0 3245.5 5887.0 RDEx-CSOP achieves the highest total score (53680.553680.5) and the best average rank (1.361.36) across all four released algorithms. It leads the full field in the combined U-score and in the Speed category, while remaining competitive in the Accuracy and Constraint categories. The earlier RDEx baseline ranks second with a total score of 49155.049155.0, confirming that the track-specific reconstruction further improves the official competition metric rather than only preserving the baseline level. I-D2 Primary Statistical Tests Besides the official U-score, we report a single feasibility-aware final-quality indicator Qp(x)=f(x),CV(x)≤0,Bp+CV(x),CV(x)>0,Q_p(x)= casesf(x),&CV(x)≤ 0,\\ B_p+CV(x),&CV(x)>0, cases (18) where BpB_p is the largest finite final objective value on problem p plus 11. This construction preserves the feasibility-first ordering and enables standard Wilcoxon and Friedman tests on one scalar per run. Table I reports pairwise Wilcoxon W/T/L, Holm-corrected W/T/L, and median Vargha–Delaney A12A_12 values for QpQ_p and T. TABLE I: Primary pairwise summary over the 28 CEC2025 CSOP functions (25 runs). 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 for feasibility-aware final quality and time-to-target (larger is better for minimization). Competitor Final Q T W/T/L Holm A12A_12 W/T/L Holm A12A_12 RDEx 1/26/1 1/27/0 0.52 4/24/0 1/27/0 0.58 UDEIII 16/4/8 15/5/8 0.84 19/3/6 19/3/6 0.89 CL-SRDE 6/18/4 6/19/3 0.50 24/4/0 23/5/0 0.92 Against the earlier RDEx baseline, RDEx-CSOP is statistically comparable on final feasibility-aware quality (1/26/11/26/1) but faster on T (4/24/04/24/0). Against UDEIII and CL-SRDE, RDEx-CSOP wins 16/4/816/4/8 and 6/18/46/18/4 functions on QpQ_p, and 19/3/619/3/6 and 24/4/024/4/0 functions on T, respectively. Table I further reports Friedman average ranks. TABLE I: Primary Friedman tests on per-function medians over the 28 CEC2025 CSOP functions (25 runs). Final Q: χ2=2.90χ^2=2.90, df=3df=3, p=0.408p=0.408; T: χ2=30.47χ^2=30.47, df=3df=3, p=2.62E−06p=2.62E-06. Lower average rank indicates better performance. Algorithm Final Q T RDEx-CSOP 2.29 1.61 RDEx 2.39 2.14 UDEIII 2.84 2.89 CL-SRDE 2.48 3.36 The Friedman test is significant for T (p=2.62×10−6p=2.62× 10^-6) but not for QpQ_p (p=0.408p=0.408), indicating that the official U-score advantage is driven mainly by faster target attainment under competitive feasibility-aware final quality. I-D3 Supplementary Diagnostics Because CSOP comparison is feasibility-first, split final-objective, split final-violation, and AUC analyses are treated only as complementary diagnostics. Appendix A therefore separates supplementary solid statistical tables (full per-function QpQ_p and T results) from a complementary-diagnostics section containing split objective, split violation, and AUC analyses. I-E Time Complexity Let D denote the problem dimension, N the current front size, and MaxFEMaxFE the evaluation budget. Let TfT_f be the average cost of evaluating both f(x)f(x) and cv(x)cv(x) once. Per generation, RDEx-CSOP performs O(ND)O(ND) arithmetic operations for variation, repair, and bookkeeping, plus typically O(NlogN)O(N N) operations for ranking and front management. Across a full run, the evaluation cost O(MaxFE⋅Tf)O(MaxFE· T_f) dominates, while the algorithmic overhead is approximately O(MaxFE⋅D)O(MaxFE· D) under the usual setting with N=O(D)N=O(D). IV Conclusion This report presented RDEx-CSOP and its evaluation on the CEC 2025 CSOP benchmark suite. The results demonstrate first-place official U-score performance against all released comparison algorithms and robust feasibility behaviour on most problems, supporting the effectiveness of combining success-history adaptation with an ε -constraint mechanism and exploitation-biased hybrid search. 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] 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, §I-C. [2] K. Deb (2000) An efficient constraint handling method for genetic algorithms. Computer Methods in Applied Mechanics and Engineering 186 (2-4), p. 311–338. Cited by: §I. [3] E. Mezura-Montes and C. A. Coello Coello (2011) Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm and Evolutionary Computation 1 (4), p. 173–194. Cited by: §I. [4] A. W. Mohamed, A. A. Hadi, and K. M. Jambi (2019) Novel mutation strategy for enhancing SHADE and LSHADE algorithms for global numerical optimization. Swarm and Evolutionary Computation 50, p. 100455. Cited by: §I, §I-D. [5] 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. [6] V. Stanovov, S. Akhmedova, and E. Semenkin (2018) LSHADE algorithm with rank-based selective pressure strategy for solving CEC 2017 benchmark problems. In 2018 IEEE Congress on Evolutionary Computation (CEC), p. 1–8. Cited by: §I. [7] V. Stanovov and E. Semenkin (2024) Adaptation of the scaling factor based on the success rate in differential evolution. Mathematics 12 (4), p. 516. Cited by: §I, §I-C. [8] V. Stanovov and E. Semenkin (2024) Success rate-based adaptive differential evolution L-SRTDE for CEC 2024 competition. In 2024 IEEE Congress on Evolutionary Computation (CEC), p. 1–8. Cited by: §I, §I-C. [9] 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. [10] T. Takahama and S. Sakai (2006) Constrained optimization by the ε constrained differential evolution with gradient-based mutation and feasible elites. In 2006 IEEE International Conference on Evolutionary Computation, p. 1–8. Cited by: §I, §I-B. [11] 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, §I-D. [12] 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. [13] R. Tanabe and A. Fukunaga (2020) Reviewing and benchmarking parameter control methods in differential evolution. IEEE Transactions on Cybernetics 50 (3), p. 1170–1184. Cited by: §I. [14] 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. Appendix A Supplementary U-score Tables TABLE IV: CEC 2025 CSOP evaluation (median target): average rankings over 28 problems (lower is better) for all released comparison algorithms. Rank Algorithm Total Rank Avg Rank/Prob. Avg Speed Avg Accuracy Avg Constraint 1 RDEx-CSOP 38.0 1.36 1.59 2.75 2.52 2 RDEx 63.0 2.25 2.20 2.30 2.52 3 UDEIII 83.0 2.96 2.89 2.48 2.41 4 CL-SRDE 96.0 3.43 3.32 2.46 2.55 Appendix B Supplementary Solid Statistical Tables TABLE V: Feasibility-aware final-quality comparison on the 28 CEC2025 CSOP functions. For each run, the final-quality score equals the final objective/IGD value for feasible runs and Bp+CVB_p+CV for infeasible runs, where BpB_p is the largest finite final objective/IGD value on problem p plus 11 (smaller is better). Problem RDEx-CSOP RDEx UDEIII CL-SRDE Mean SD Mean SD W Mean SD W Mean SD W 1 3.27E-30 4.75E-30 5.92E-30 9.27E-30 = 1.57E-28 9.70E-29 + 1.26E-31 6.18E-31 = 2 3.74E-30 5.62E-30 3.04E-30 4.86E-30 = 1.31E-28 7.83E-29 + 7.24E-31 2.42E-30 = 3 2.56E+02 7.36E+01 3.39E+02 1.63E+02 = 9.22E+01 5.54E+01 - 5.37E+02 1.07E+02 + 4 1.54E+01 1.42E+00 1.57E+01 1.45E+00 = 6.51E+00 6.78E+00 - 4.84E+01 8.86E+00 + 5 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 1.17E-28 4.17E-28 = 0.00E+00 0.00E+00 = 6 0.00E+00 0.00E+00 2.29E+00 1.00E+01 = 0.00E+00 0.00E+00 = 2.87E+00 1.41E+01 = 7 -1.39E+03 6.80E+02 -8.70E+02 1.54E+02 + -6.73E+02 1.49E+02 + -8.28E+02 1.80E+02 + 8 -2.84E-04 0.00E+00 -2.84E-04 0.00E+00 = -2.84E-04 5.85E-12 + -2.84E-04 0.00E+00 = 9 -2.67E-03 0.00E+00 -2.67E-03 0.00E+00 = -2.67E-03 4.34E-19 + -2.67E-03 0.00E+00 = 10 -1.03E-04 0.00E+00 -1.03E-04 0.00E+00 = -1.03E-04 1.36E-20 + -1.03E-04 0.00E+00 = 11 9.04E+00 3.65E+00 5.90E+00 5.49E+00 - 5.25E+01 1.23E+02 + 5.44E-02 3.05E+00 - 12 9.54E+00 1.14E+00 9.78E+00 0.00E+00 = 3.99E+00 2.18E-02 - 1.01E+01 4.05E+00 = 13 9.34E-29 2.14E-28 1.17E-28 2.34E-28 = 4.78E-01 1.30E+00 + 4.67E-29 1.58E-28 = 14 1.41E+00 0.00E+00 1.42E+00 2.36E-02 = 1.41E+00 0.00E+00 - 1.47E+00 4.05E-02 + 15 -3.93E+00 4.44E-16 -3.93E+00 1.18E-05 = 2.36E+00 1.41E-06 + -3.42E+00 1.15E+00 = 16 1.99E+01 4.09E+00 2.10E+01 3.69E+00 = 0.00E+00 0.00E+00 - 2.32E+01 3.47E+00 + 17 3.35E+01 0.00E+00 3.35E+01 0.00E+00 = 3.19E+01 7.89E-01 - 3.35E+01 0.00E+00 = 18 3.65E+01 0.00E+00 3.65E+01 1.96E-05 = 8.00E+03 2.49E+03 + 3.65E+01 0.00E+00 = 19 4.27E+04 0.00E+00 4.27E+04 0.00E+00 = 4.28E+04 7.28E-12 + 4.27E+04 0.00E+00 = 20 1.26E+00 2.11E-01 1.33E+00 2.46E-01 = 1.85E+00 2.82E-01 + 2.43E+00 6.99E-01 + 21 2.16E+01 8.90E+00 2.39E+01 7.92E+00 = 9.28E+00 8.34E+00 - 2.47E+01 1.54E+01 = 22 3.75E-26 1.49E-26 5.61E-26 3.84E-26 = 2.56E+01 4.84E+01 + 2.85E-26 6.52E-27 - 23 1.41E+00 0.00E+00 1.41E+00 0.00E+00 = 1.45E+00 4.34E-02 = 1.41E+00 0.00E+00 = 24 -3.93E+00 4.44E-16 -3.93E+00 4.44E-16 = 2.36E+00 9.80E-08 + -3.93E+00 4.44E-16 = 25 2.25E+01 3.00E+00 2.36E+01 3.69E+00 = 2.51E-01 1.23E+00 - 2.46E+01 3.10E+00 = 26 3.30E+01 0.00E+00 3.30E+01 0.00E+00 = 3.26E+01 8.09E-01 = 3.30E+01 0.00E+00 = 27 3.65E+01 8.06E-05 3.65E+01 8.00E-05 = 1.46E+04 5.11E+03 + 3.65E+01 2.71E-05 - 28 4.29E+04 1.14E+01 4.29E+04 1.10E+01 = 4.30E+04 1.47E+01 + 4.29E+04 7.28E-12 - W/T/L −/−/−-/-/- 1/26/1 16/4/8 6/18/4 TABLE VI: Time-to-target comparison on the 28 CEC2025 CSOP functions. T is the first checkpoint index where the run reaches the median target (smaller is better); runs that never reach the target are assigned 2001. Problem RDEx-CSOP RDEx UDEIII CL-SRDE Mean SD Mean SD W Mean SD W Mean SD W 1 1332.6 546.2 1382.6 548.3 = 1979.6 105.0 + 1327.4 137.8 = 2 1251.2 514.8 1298.5 527.0 = 1960.7 138.2 + 1399.3 222.3 + 3 1575.3 541.4 1813.9 272.3 = 394.8 162.1 - 2001.0 0.0 + 4 1607.4 529.8 1569.9 496.5 = 837.6 50.4 - 2001.0 0.0 + 5 986.2 12.0 986.6 13.9 = 1476.4 466.1 + 1395.2 14.8 + 6 577.4 10.3 807.2 521.1 = 746.3 37.6 + 991.0 206.4 + 7 334.7 369.4 1130.0 739.4 + 1906.6 320.2 + 1606.0 480.3 + 8 765.0 13.3 769.6 11.7 = 2001.0 0.0 + 1221.0 7.6 + 9 50.5 12.9 56.2 12.3 = 1322.4 905.8 + 111.6 26.5 + 10 793.0 12.9 793.3 11.6 = 2001.0 0.0 + 1231.7 9.0 + 11 1.7 0.6 1.5 0.6 = 1.7 0.8 = 1.9 1.1 = 12 574.1 258.7 580.4 150.6 = 269.9 33.2 - 843.6 257.9 + 13 1172.4 361.7 1216.1 392.6 = 1672.0 383.1 + 1484.0 152.8 + 14 421.9 55.6 541.9 431.8 = 1036.1 2.7 + 1618.0 600.7 + 15 59.9 13.7 70.0 13.2 + 2001.0 0.0 + 148.7 25.8 + 16 1243.6 732.8 1468.6 667.3 = 131.8 71.2 - 1926.3 212.1 + 17 842.0 807.3 1232.2 869.8 = 212.0 197.6 - 1603.7 643.4 + 18 200.1 8.3 201.6 6.0 = 2001.0 0.0 + 307.7 7.1 + 19 280.2 25.9 290.6 40.3 = 311.4 49.3 + 560.7 254.5 + 20 1078.6 444.5 1084.2 472.0 = 1965.8 76.0 + 1922.5 215.4 + 21 396.5 173.5 498.9 341.1 = 464.7 455.1 = 800.5 556.8 + 22 1525.7 421.6 1568.4 415.9 = 2001.0 0.0 + 1560.4 130.4 = 23 433.6 28.0 452.5 30.1 + 1851.7 202.0 + 708.2 13.4 + 24 60.7 14.2 68.6 13.6 = 2001.0 0.0 + 141.8 21.9 + 25 1315.1 782.3 1459.0 727.9 = 515.5 424.5 - 1857.0 304.0 = 26 1102.0 803.1 1206.8 806.3 = 826.8 873.9 = 1772.2 460.9 + 27 218.8 34.6 216.3 13.5 = 2001.0 0.0 + 324.0 7.7 + 28 263.0 12.8 272.4 11.6 + 2001.0 0.0 + 459.6 21.2 + W/T/L −/−/−-/-/- 4/24/0 19/3/6 24/4/0 Appendix C Complementary Diagnostics The tables in this section are diagnostic supplements to the official U-score results. For constrained tracks, split objective/IGD and split final-CV comparisons are not treated as independent primary criteria. TABLE VII: Complementary pairwise summary over the 28 CEC2025 CSOP functions (25 runs). For each metric (Final Obj., Final CV, 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 Obj. Final CV AUC W/T/L Holm A12A_12 W/T/L Holm A12A_12 W/T/L Holm A12A_12 RDEx 1/27/0 1/27/0 0.54 1/26/1 0/28/0 0.50 1/27/0 1/27/0 0.55 UDEIII 14/7/7 14/7/7 0.73 22/5/1 21/6/1 1.00 15/4/9 14/6/8 0.73 CL-SRDE 10/16/2 10/17/1 0.51 2/22/4 2/22/4 0.50 28/0/0 28/0/0 1.00 TABLE VIII: Complementary Friedman tests on per-function medians over the 28 CEC2025 CSOP functions (25 runs). Final Obj.: χ2=4.25χ^2=4.25, df=3df=3, p=0.234p=0.234; Final CV: χ2=29.00χ^2=29.00, df=3df=3, p=4.82E−06p=4.82E-06; AUC: χ2=39.13χ^2=39.13, df=3df=3, p=7.74E−08p=7.74E-08. Lower average rank indicates better performance for each metric. Algorithm Final Obj. Final CV AUC RDEx-CSOP 2.11 2.21 1.75 RDEx 2.46 2.20 1.96 UDEIII 2.66 3.62 2.57 CL-SRDE 2.77 1.96 3.71 TABLE IX: Final objective comparison on the 28 CEC2025 CSOP functions. For each algorithm, the mean and SD over 25 runs are reported; W gives the Wilcoxon outcome of RDEx-CSOP against the competitor. Problem RDEx-CSOP RDEx UDEIII CL-SRDE Mean SD Mean SD W Mean SD W Mean SD W 1 3.27E-30 4.75E-30 5.92E-30 9.27E-30 = 1.57E-28 9.70E-29 + 1.26E-31 6.18E-31 = 2 3.74E-30 5.62E-30 3.04E-30 4.86E-30 = 1.31E-28 7.83E-29 + 7.24E-31 2.42E-30 = 3 2.56E+02 7.36E+01 3.39E+02 1.63E+02 = 9.22E+01 5.54E+01 - 5.37E+02 1.07E+02 + 4 1.54E+01 1.42E+00 1.57E+01 1.45E+00 = 6.51E+00 6.78E+00 - 4.84E+01 8.86E+00 + 5 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 1.17E-28 4.17E-28 = 0.00E+00 0.00E+00 = 6 0.00E+00 0.00E+00 2.29E+00 1.00E+01 = 0.00E+00 0.00E+00 = 2.87E+00 1.41E+01 = 7 -1.39E+03 6.80E+02 -8.70E+02 1.54E+02 + -6.73E+02 1.49E+02 + -8.28E+02 1.80E+02 + 8 -2.84E-04 0.00E+00 -2.84E-04 0.00E+00 = -2.84E-04 5.85E-12 + -2.84E-04 0.00E+00 = 9 -2.67E-03 0.00E+00 -2.67E-03 0.00E+00 = -2.67E-03 4.34E-19 + -2.67E-03 0.00E+00 = 10 -1.03E-04 0.00E+00 -1.03E-04 0.00E+00 = -1.03E-04 1.36E-20 + -1.03E-04 0.00E+00 = 11 -7.87E+00 5.44E+00 -5.09E+00 4.57E+00 = -1.98E+02 4.90E+02 = -1.59E+00 2.54E+00 + 12 9.54E+00 1.14E+00 9.78E+00 0.00E+00 = 3.99E+00 2.18E-02 - 1.01E+01 4.05E+00 = 13 9.34E-29 2.14E-28 1.17E-28 2.34E-28 = 4.78E-01 1.30E+00 + 4.67E-29 1.58E-28 = 14 1.41E+00 0.00E+00 1.42E+00 2.36E-02 = 1.41E+00 0.00E+00 - 1.47E+00 4.05E-02 + 15 -3.93E+00 4.44E-16 -3.93E+00 1.18E-05 = 2.36E+00 1.41E-06 + -3.42E+00 1.15E+00 = 16 1.99E+01 4.09E+00 2.10E+01 3.69E+00 = 0.00E+00 0.00E+00 - 2.32E+01 3.47E+00 + 17 7.54E-01 1.46E-01 8.37E-01 9.42E-02 = 6.81E-01 4.33E-01 = 8.41E-01 8.11E-02 + 18 3.65E+01 0.00E+00 3.65E+01 1.96E-05 = 1.85E+02 2.89E+01 + 3.65E+01 0.00E+00 = 19 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 20 1.26E+00 2.11E-01 1.33E+00 2.46E-01 = 1.85E+00 2.82E-01 + 2.43E+00 6.99E-01 + 21 2.16E+01 8.90E+00 2.39E+01 7.92E+00 = 9.28E+00 8.34E+00 - 2.47E+01 1.54E+01 = 22 3.75E-26 1.49E-26 5.61E-26 3.84E-26 = 2.56E+01 4.84E+01 + 2.85E-26 6.52E-27 - 23 1.41E+00 0.00E+00 1.41E+00 0.00E+00 = 1.45E+00 4.34E-02 = 1.41E+00 0.00E+00 = 24 -3.93E+00 4.44E-16 -3.93E+00 4.44E-16 = 2.36E+00 9.80E-08 + -3.93E+00 4.44E-16 = 25 2.25E+01 3.00E+00 2.36E+01 3.69E+00 = 2.51E-01 1.23E+00 - 2.46E+01 3.10E+00 = 26 7.53E-01 1.50E-01 8.11E-01 7.52E-02 = 6.89E-01 2.63E-01 = 8.56E-01 1.18E-01 + 27 3.65E+01 8.06E-05 3.65E+01 8.00E-05 = 2.89E+02 3.21E+01 + 3.65E+01 2.71E-05 - 28 -5.12E+00 2.97E+00 -4.75E+00 3.05E+00 = 7.72E+01 1.60E+01 + 0.00E+00 0.00E+00 + W/T/L −/−/−-/-/- 1/27/0 14/7/7 10/16/2 TABLE X: Final constraint-violation comparison on the 28 CEC2025 CSOP functions. For each algorithm, the mean and SD over 25 runs are reported; W gives the Wilcoxon outcome of RDEx-CSOP against the competitor. Problem RDEx-CSOP RDEx UDEIII CL-SRDE Mean SD Mean SD W Mean SD W Mean SD W 1 -2.70E+05 0.00E+00 -2.70E+05 0.00E+00 = 0.00E+00 0.00E+00 + -2.70E+05 0.00E+00 = 2 -2.70E+05 0.00E+00 -2.70E+05 0.00E+00 = 0.00E+00 0.00E+00 + -2.70E+05 0.00E+00 = 3 -1.99E+05 1.85E+04 -1.85E+05 1.90E+04 + 0.00E+00 0.00E+00 + -1.52E+05 2.11E+04 + 4 -4.45E+00 3.34E+00 -5.01E+00 3.12E+00 = 0.00E+00 0.00E+00 + -2.01E+01 3.74E+00 - 5 -2.14E+03 0.00E+00 -2.14E+03 0.00E+00 = 0.00E+00 0.00E+00 + -2.14E+03 0.00E+00 = 6 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 7 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 8 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 9 -6.05E-37 2.80E-37 -4.91E-37 3.04E-37 = 0.00E+00 0.00E+00 + -4.96E-37 3.07E-37 = 10 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 11 6.00E-10 1.51E-09 9.74E-11 4.67E-10 - 4.51E+01 1.22E+02 + 2.40E-15 1.16E-14 - 12 -7.64E-01 1.49E-01 -7.94E-01 4.80E-05 = 0.00E+00 0.00E+00 + -7.67E-01 2.70E-01 = 13 -1.25E+02 0.00E+00 -1.25E+02 0.00E+00 = 0.00E+00 0.00E+00 + -1.25E+02 0.00E+00 = 14 -1.00E+00 1.11E-16 -9.20E-01 2.71E-01 = 0.00E+00 0.00E+00 + -3.20E-01 4.66E-01 + 15 -1.37E+02 1.89E+02 -1.73E+02 1.92E+02 = 0.00E+00 0.00E+00 + -4.83E+02 2.65E+02 - 16 -2.97E+03 1.02E+01 -2.97E+03 9.87E+00 = 0.00E+00 0.00E+00 + -2.97E+03 1.12E+01 = 17 3.10E+01 0.00E+00 3.10E+01 0.00E+00 = 2.94E+01 7.89E-01 - 3.10E+01 0.00E+00 = 18 -3.00E+03 4.55E-13 -3.00E+03 4.55E-13 = 7.75E+03 2.49E+03 + -3.00E+03 4.55E-13 = 19 4.27E+04 0.00E+00 4.27E+04 0.00E+00 = 4.27E+04 7.28E-12 + 4.27E+04 0.00E+00 = 20 -3.66E-01 1.81E-01 -2.89E-01 1.81E-01 = 0.00E+00 0.00E+00 + -3.87E-01 1.73E-01 = 21 -1.32E+00 3.97E-01 -1.42E+00 3.53E-01 = 0.00E+00 0.00E+00 + -1.33E+00 6.25E-01 = 22 -1.25E+02 0.00E+00 -1.25E+02 0.00E+00 = 0.00E+00 0.00E+00 + -1.25E+02 0.00E+00 = 23 -1.00E+00 1.11E-16 -1.00E+00 1.11E-16 = 0.00E+00 0.00E+00 + -1.00E+00 1.11E-16 = 24 -7.38E+01 1.00E+02 -8.14E+01 1.36E+02 = 0.00E+00 0.00E+00 + -1.98E+02 2.32E+02 = 25 -2.97E+03 8.21E+00 -2.97E+03 1.04E+01 = 0.00E+00 0.00E+00 + -2.97E+03 8.66E+00 = 26 3.10E+01 0.00E+00 3.10E+01 0.00E+00 = 3.06E+01 8.09E-01 = 3.10E+01 0.00E+00 = 27 -3.00E+03 4.55E-13 -3.00E+03 4.55E-13 = 1.43E+04 5.11E+03 + -3.00E+03 4.55E-13 = 28 4.28E+04 1.14E+01 4.28E+04 1.10E+01 = 4.29E+04 1.47E+01 + 4.27E+04 0.00E+00 - W/T/L −/−/−-/-/- 1/26/1 22/5/1 2/22/4 TABLE XI: Anytime convergence comparison using AUC over 2000 checkpoints on the 28 CEC2025 CSOP 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-CSOP RDEx UDEIII CL-SRDE Mean SD Mean SD W Mean SD W Mean SD W 1 0.17 0.01 0.17 0.00 = 0.14 0.01 - 0.32 0.01 + 2 0.20 0.01 0.20 0.01 = 0.14 0.01 - 0.38 0.01 + 3 1.87 0.65 2.18 0.50 = 0.50 0.18 - 3.15 0.11 + 4 0.78 0.24 0.84 0.32 = 0.81 0.04 = 1.80 0.08 + 5 0.36 0.01 0.36 0.01 = 0.30 0.01 - 0.60 0.02 + 6 0.27 0.01 0.39 0.33 = 0.52 0.02 + 0.62 0.30 + 7 0.74 0.66 1.34 0.86 + 2.30 0.47 + 2.00 0.61 + 8 0.10 0.00 0.10 0.00 = 0.33 0.06 + 0.20 0.01 + 9 0.02 0.00 0.02 0.00 = 0.06 0.04 + 0.06 0.01 + 10 0.10 0.00 0.10 0.00 = 0.39 0.05 + 0.19 0.00 + 11 0.02 0.04 0.04 0.05 = 0.04 0.06 = 0.10 0.03 + 12 0.30 0.01 0.30 0.01 = 0.26 0.03 - 0.56 0.20 + 13 0.70 0.02 0.70 0.02 = 0.59 0.21 - 1.16 0.04 + 14 0.10 0.01 0.10 0.01 = 0.11 0.02 + 0.30 0.10 + 15 0.06 0.01 0.06 0.01 = 0.90 0.02 + 0.30 0.17 + 16 0.54 0.33 0.63 0.32 = 0.12 0.07 - 1.06 0.18 + 17 0.02 0.01 0.03 0.02 = 0.05 0.04 = 0.05 0.01 + 18 0.27 0.02 0.27 0.01 = 2.18 0.06 + 0.43 0.01 + 19 0.23 0.01 0.24 0.02 = 0.09 0.03 - 0.37 0.01 + 20 0.33 0.04 0.34 0.04 = 0.39 0.05 + 0.61 0.06 + 21 0.32 0.03 0.32 0.02 = 0.42 0.21 + 0.70 0.40 + 22 0.82 0.04 0.83 0.04 = 1.51 0.46 + 1.35 0.03 + 23 0.11 0.00 0.12 0.00 = 0.19 0.02 + 0.19 0.00 + 24 0.09 0.01 0.09 0.02 = 0.91 0.02 + 0.26 0.02 + 25 0.55 0.23 0.63 0.26 = 0.43 0.40 - 1.01 0.15 + 26 0.03 0.01 0.04 0.01 = 0.03 0.02 = 0.08 0.02 + 27 0.31 0.03 0.31 0.02 = 2.43 0.04 + 0.48 0.01 + 28 0.20 0.02 0.20 0.02 = 1.98 0.06 + 0.38 0.01 + W/T/L −/−/−-/-/- 1/27/0 15/4/9 28/0/0