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RDEx-SOP: Exploitation-Biased Reconstructed Differential Evolution for Fixed-Budget Bound-Constrained Single-Objective Optimization
Sichen Tao, Yifei Yang, Ruihan Zhao, Kaiyu Wang, Sicheng Liu, Shangce Gao
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Summary
RDEx-SOP is an exploitation-biased differential evolution algorithm designed for fixed-budget, bound-constrained single-objective optimization, specifically for the IEEE CEC 2025 competition. It integrates success-history parameter adaptation, a two-branch mutation mechanism (standard and exploitation-biased), and Cauchy local perturbation to optimize the speed-accuracy trade-off under the U-score framework.
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RDEx-SOP → competesin → IEEE CEC 2025
confidence 100% · RDEx-SOP, an exploitation-biased success-history differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition
RDEx-SOP → isa → Differential Evolution
confidence 100% · RDEx-SOP: Exploitation-Biased Reconstructed Differential Evolution
RDEx-SOP → usesframework → U-score
confidence 100% · We evaluate RDEx-SOP on the official CEC 2025 SOP benchmark with the U-score framework
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Abstract
Abstract:Bound-constrained single-objective numerical optimisation remains a key benchmark for assessing the robustness and efficiency of evolutionary algorithms. This report documents RDEx-SOP, an exploitation-biased success-history differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session). RDEx-SOP combines success-history parameter adaptation, an exploitation-biased hybrid branch, and lightweight local perturbations to balance fast convergence and final solution quality under a strict evaluation budget. We evaluate RDEx-SOP on the official CEC 2025 SOP benchmark with the U-score framework (Speed and Accuracy categories). Experimental results show that RDEx-SOP achieves strong overall performance and statistically competitive final outcomes across the 29 benchmark functions.
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- Source: https://arxiv.org/abs/2603.27089v1
- Canonical: https://arxiv.org/abs/2603.27089v1
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RDEx-SOP: Exploitation-Biased Reconstructed Differential Evolution for Fixed-Budget Bound-Constrained Single-Objective Optimization Sichen Tao1,2, Yifei Yang3, Ruihan Zhao4,5, Kaiyu Wang6,7, Sicheng Liu8, Shangce Gao1 Abstract Bound-constrained single-objective numerical optimisation remains a key benchmark for assessing the robustness and efficiency of evolutionary algorithms. This report documents RDEx-SOP, an exploitation-biased success-history differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session). RDEx-SOP combines success-history parameter adaptation, an exploitation-biased hybrid branch, and lightweight local perturbations to balance fast convergence and final solution quality under a strict evaluation budget. We evaluate RDEx-SOP on the official CEC 2025 SOP benchmark with the U-score framework (Speed and Accuracy categories). Experimental results show that RDEx-SOP achieves strong overall performance and statistically competitive final outcomes across the 29 benchmark functions. Index Terms: Differential Evolution, Numerical Optimisation, CEC 2025, Single-objective, Bound-constrained, U-score I Introduction Differential evolution (DE) remains one of the most competitive paradigms for continuous black-box optimisation because its search directions are generated directly from population differences, which gives strong landscape adaptivity at relatively low algorithmic complexity [9]. In single-objective bound-constrained optimisation, the most successful DE variants have typically emerged not from completely new frameworks, but from a sustained sequence of refinements to mutation bias, parameter adaptation, and population scheduling. This development line is now fairly clear. JADE introduced the current-to-ppbest mutation and archive mechanism that became the backbone of many later DE variants [13]. SHADE then replaced manually tuned parameters with success-history memories, and L-SHADE further improved the line by coupling success-history adaptation with linear population size reduction [11, 10]. Subsequent variants such as iL-SHADE and jSO refined parameter control and mutation pressure, while LSHADE-RSP and iLSHADE-RSP strengthened selective pressure and local perturbation, respectively [1, 2, 6, 3]. This progression also supports the broader conclusion that success-history control is one of the most effective and best-benchmarked parameter-control directions in DE [12]. Two more recent observations are especially relevant to competition settings. First, ordered mutation can deliberately increase exploitation pressure without discarding differential information, as demonstrated by EB-LSHADE [4]. Second, the success rate itself can serve as a compact learning signal for scaling-factor control, as shown in the recent LSRTDE line [7, 8]. However, these ingredients are usually studied in isolation, and many comparisons still emphasise final objective values more than the joint speed-accuracy trade-off that matters in fixed-budget competitions. The CEC 2025 SOP track makes that trade-off explicit because algorithms are ranked under the U-score framework, which rewards both target-attainment speed and final solution quality under a fixed evaluation budget [5]. RDEx-SOP is designed for this setting as a reconstructed DE that selectively recombines the most effective ideas from the post-SHADE literature rather than proposing a single isolated operator. Its main contribution is a compact competition-oriented formulation that couples success-history memories, linear population reduction, exploitation-biased hybrid mutation, adaptive operator allocation, and lightweight Cauchy local perturbation, together with an empirical study showing that this reconstruction is highly competitive on the official CEC 2025 benchmark. The source code for RDEx-SOP is publicly available on Sichen Tao’s GitHub page: https://github.com/SichenTao. I Reconstructed Differential Evolution with Exploitation-biased Hybridization (RDEx-SOP) I-A Problem Formulation The bound-constrained single-objective optimisation problem considered in the SOP track can be written as: minx∈ℝDf(x),s.t. ℓj≤xj≤uj,j=1,2,…,D. _x ^Df(x), .t. _j≤ x_j≤ u_j,\;j=1,2,…,D. (1) I-B Overall Framework RDEx-SOP is a success-history adaptive differential evolution method with three main design elements: • an elite front population with linear size reduction [10]; • a two-branch mutation mechanism (standard branch and exploitation-biased (EB) branch) with an adaptive hybrid rate, following the ordered-mutation intuition of EB-LSHADE [4]; • a light local perturbation operator based on the Cauchy distribution [3]. Let ℱ(g)=xi(g)i=1N(g)F^(g)=\x_i^(g)\_i=1^N^(g) denote the front at generation g, where N(g)N^(g) is the current front size. For each target vector xi(g)x_i^(g), RDEx-SOP samples control parameters (Fi(g),CRi(g))(F_i^(g),CR_i^(g)), generates a donor vector vi(g)v_i^(g) using one of the two mutation branches, and then applies binomial crossover to obtain a trial ui(g)u_i^(g). If f(ui(g))≤f(xi(g))f(u_i^(g))≤ f(x_i^(g)), the trial is accepted and contributes to the success-history update. I-C Standard Branch Mutation and Crossover I-C1 Mutation In the standard branch, RDEx-SOP uses a current-to-ppbest/1-like operator with an additional difference vector: 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) ), (2) where xpbest(g)x_pbest^(g) is selected from the top p(g)p^(g) individuals of ℱ(g)F^(g), and xr1(g)x_r_1^(g), xr2(g)x_r_2^(g) are randomly selected vectors (with indices different from each other and from i). The selection window size p(g)p^(g) is dynamically adjusted according to the empirical success rate: p(g)=max(2,⌊N(g)⋅ξ⋅exp(−k⋅SR(g))⌋),p^(g)= (2, N^(g)·ξ· (-k· SR^(g) ) ), (3) where SR(g)∈[0,1]SR^(g)∈[0,1] is the fraction of successful trials within the front in generation g, and (ξ,k)(ξ,k) are constants. I-C2 Binomial Crossover and Bound Handling RDEx-SOP applies binomial crossover to combine vi(g)v_i^(g) and xi(g)x_i^(g): ui,j(g)=vi,j(g),if rand(0,1)<CRi(g)or j=jrand,xi,j(g),otherwise,u^(g)_i,j= casesv^(g)_i,j,&if rand(0,1)<CR_i^(g)\ or j=j_rand,\\ x^(g)_i,j,&otherwise, cases (4) where jrand∈1,…,Dj_rand∈\1,…,D\ guarantees that at least one dimension is inherited from the donor. If a component violates the bounds, RDEx-SOP repairs it by resampling uniformly within [ℓj,uj][ _j,u_j]. I-D Exploitation-biased (EB) Branch The EB branch uses an ordered donor set to bias the search toward fitter regions while keeping a differential term for diversity. Let xa(g),xb(g),xc(g)\x_a^(g),x_b^(g),x_c^(g)\ be a donor set (sampled similarly to xpbest(g),xr1(g),xr2(g)\x_pbest^(g),x_r_1^(g),x_r_2^(g)\), and let xbest(g),xmid(g),xworst(g)\x_best^(g),x_mid^(g),x_worst^(g)\ denote these three donors ordered by fitness (best = lowest objective). The EB mutation 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) ). (5) RDEx-SOP maintains an adaptive hybrid rate ρEB(g)∈[0,1] _EB^(g)∈[0,1] that controls how often the EB branch is used. After each generation, ρEB(g) _EB^(g) is updated according to the relative fitness improvements contributed by EB-generated trials versus standard-branch trials: ρEB(g+1)=∑i∈EB(g)Δfi∑i∈EB(g)Δfi+∑i∈std(g)Δfi, _EB^(g+1)= _i _EB^(g) f_i _i _EB^(g) f_i+ _i _std^(g) f_i, (6) where Δfi=f(xi(g))−f(ui(g)) f_i=f(x_i^(g))-f(u_i^(g)) and EB(g)S_EB^(g) and std(g)S_std^(g) are the sets of successful trials generated by the EB and standard branches, respectively. I-E Success-history Parameter Sampling and Update RDEx-SOP uses two memories, MFM_F and MCRM_CR, each with H entries, to store reference values of successful parameters. In each generation, a memory index is selected and perturbed to generate (Fi(g),CRi(g))(F_i^(g),CR_i^(g)). In the standard branch, Fi(g)F_i^(g) is sampled from a truncated Gaussian distribution, whose mean depends on the current success rate in the spirit of recent success-rate-driven DE adaptation [7]: μF(g)=0.4+0.25⋅tanh(5⋅SR(g)), _F^(g)=0.4+0.25· (5· SR^(g) ), (7) and CRi(g)CR_i^(g) is sampled from a truncated Gaussian distribution centered at the memory value. In the EB branch, Fi(g)F_i^(g) is sampled from a Cauchy distribution centered at the memory value (with a fallback reference value when needed), and CRi(g)CR_i^(g) is sampled similarly with early-stage lower bounds to encourage larger crossovers. After the generation, successful parameters update the memory via a weighted Lehmer mean. Let Fkk=1|(g)|\F_k\_k=1^|S^(g)| and CRkk=1|(g)|\CR_k\_k=1^|S^(g)| denote parameters of successful trials, and let the weights be wk=Δfk/∑tΔftw_k= f_k/ _t f_t. Then a memory entry is updated as: MF←∑kwkFk2∑kwkFk,M_F← _kw_kF_k^2 _kw_kF_k, (8) MCR←12(MCR+∑kwkCRk2∑kwkCRk).M_CR← 12 (M_CR+ _kw_kCR_k^2 _kw_kCR_k ). (9) I-F Cauchy Local Perturbation To encourage local refinement (especially when crossover does not modify certain coordinates), RDEx-SOP applies a Cauchy perturbation to non-crossover dimensions with probability prp_r: ui,j(g)←Cauchy(xi,j(g),σloc).u^(g)_i,j (x^(g)_i,j, _loc ). (10) This operator introduces occasional heavy-tailed jumps while keeping most coordinates unchanged. I-G Linear Population Size Reduction Following the L-SHADE line [10], RDEx-SOP linearly reduces the front size from N0N_0 to a small minimum value NminN_ as the evaluation budget is consumed: N(g+1)=⌊N0+(Nmin−N0)⋅NFEMaxFE⌋.N^(g+1)= N_0+(N_ -N_0)· NFEMaxFE . (11) I-H Pseudocode Algorithm 1 summarizes the overall RDEx-SOP procedure. Input: Population size N0N_0, memory size H, evaluation budget MaxFEMaxFE. Output: Best solution found. 1 Initialise N(0)←N0N^(0)← N_0 and sample an initial front population ℱ(0)F^(0); 2 Evaluate all individuals and initialise memories MFM_F and MCRM_CR; 3 while NFE<MaxFENFE<MaxFE do 4 foreach xi∈ℱ(g)x_i ^(g) do 5 Sample (Fi,CRi)(F_i,CR_i) from success-history memories; 6 Select EB branch with probability ρEB(g) _EB^(g); 7 Generate donor viv_i via Eq. (2) or Eq. (5); 8 Generate trial uiu_i by crossover Eq. (4) and apply repairs/perturbation; 9 Evaluate f(ui)f(u_i) and perform greedy selection; 10 Store successful (Fi,CRi,Δfi)(F_i,CR_i, f_i); 11 12 Update ρEB _EB by Eq. (6) and update MF,MCRM_F,M_CR by Eq. (8)–(9); 13 Reduce front size using Eq. (11); 14 g←g+1g← g+1; 15 Algorithm 1 RDEx-SOP framework. I Experimental Results I-A Benchmark Functions The CEC 2025 SOP benchmark suite consists of 29 bound-constrained test functions with dimension D=30D=30 and default variable bounds [−100,100]D[-100,100]^D. Following the official competition protocol, each function is evaluated with 25 independent runs. The maximum number of function evaluations is set to MaxFE=10000×DMaxFE=10000× D, and the platform records the best-so-far objective values at 10001000 evenly spaced checkpoints (i.e., every 10×D10× D evaluations) for convergence-speed evaluation. I-B Parameter Settings Unless otherwise stated, we use the default parameter configuration of the reference RDEx-SOP implementation: • initial population/front size N0=600N_0=600 and minimum size Nmin=4N_ =4; • success-history memory size H=5H=5; • initial EB hybrid rate ρEB(0)=0.7 _EB^(0)=0.7; • local Cauchy perturbation probability pr=0.1p_r=0.1 and scale σloc=0.1 _loc=0.1; • Gaussian perturbation scale for the standard-branch F: σF=0.02 _F=0.02; • selection-pressure parameters (ξ,k)=(0.7,7.0)(ξ,k)=(0.7,7.0) in Eq. (3). I-C Experimental Settings RDEx-SOP is evaluated with the official U-score framework using the median target setting. The U-score evaluation consists of two categories: Speed (performance over checkpoints) and Accuracy (final-stage performance). We compare RDEx-SOP with a selected set of representative competitors available in this repository: LSRTDE, jSOa, IEACOP, mLSHADE-LR, and RDE. I-D Statistical Results I-D1 Overall U-score Results Table I reports the overall U-score results for the selected algorithm set. TABLE I: CEC 2025 SOP evaluation (median target): overall scores over 29 problems and 25 runs for the selected algorithm set. Rank Algorithm Total Score Avg Score/Prob. Speed Accuracy 1 RDEx 81229.5 2801.02 76490.5 4739.0 2 LSRTDE 73664.5 2540.16 68007.5 5657.0 3 RDE 69459.0 2395.14 58066.0 11393.0 4 jSOa 44653.0 1539.76 20984.0 23669.0 5 mLSHADE_LR 44106.5 1520.91 25274.5 18832.0 6 IEACOP 15312.5 528.02 5635.5 9677.0 RDEx-SOP achieves the highest total score and the best overall rank mainly due to its leading performance in the Speed category, which is consistent with the design goal of accelerating convergence through the EB hybrid branch. I-D2 Per-function Statistics and Wilcoxon Tests Due to the strict 4-page submission limit, detailed per-function statistics (means/SD and per-function Wilcoxon tests) are provided in the supplementary appendix of the full report. In terms of per-function Wilcoxon wins/ties/losses on final values, RDEx-SOP obtains 4/24/1 against LSRTDE, 21/5/3 against jSOa, and 27/1/1 against IEACOP, indicating that its final-stage accuracy is comparable to strong baselines while showing consistent improvements over several competitors. To make the pairwise comparisons more rigorous and informative, Table I further reports Holm-corrected W/T/L counts across the 29 functions (to control the family-wise error rate) and the Vargha–Delaney A12A_12 effect size for Final, T, and AUC (larger is better for minimization). TABLE I: Pairwise summary over the 29 CEC2025 SOP functions (25 runs). For each metric (Final, T, and AUC), we report uncorrected per-function Wilcoxon W/T/L at α=0.05α=0.05, Holm-corrected W/T/L across the 29 functions, and the median Vargha–Delaney A12A_12 effect size (larger is better for minimization). Competitor Final T AUC W/T/L Holm A12A_12 W/T/L Holm A12A_12 W/T/L Holm A12A_12 LSRTDE 4/24/1 2/27/0 0.48 17/12/0 10/19/0 0.73 23/6/0 21/8/0 0.79 jSOa 21/5/3 21/5/3 0.92 23/3/3 23/3/3 0.98 27/2/0 27/2/0 1.00 IEACOP 27/1/1 27/1/1 1.00 25/3/1 25/3/1 1.00 23/0/6 23/0/6 1.00 mLSHADE-LR 21/4/4 21/4/4 0.91 23/3/3 23/3/3 0.98 23/2/4 23/2/4 1.00 RDE 17/6/6 17/6/6 0.83 18/2/9 18/2/9 0.93 19/2/8 19/2/8 1.00 As complementary multi-algorithm comparisons, we report Friedman tests on per-function medians for final values, T, and AUC, together with the resulting average ranks (Table I). TABLE I: Friedman tests on per-function medians over the 29 CEC2025 SOP functions (25 runs). Final: χ2=77.67χ^2=77.67, df=5df=5, p=2.57E−15p=2.57E-15; T: χ2=63.85χ^2=63.85, df=5df=5, p=1.94E−12p=1.94E-12; AUC: χ2=60.07χ^2=60.07, df=5df=5, p=1.17E−11p=1.17E-11. Lower average rank indicates better performance for each metric. Algorithm Final T AUC RDEx-SOP 2.26 1.84 1.76 LSRTDE 2.29 2.69 2.72 jSOa 3.79 4.43 4.97 IEACOP 5.83 4.95 4.03 mLSHADE-LR 4.07 4.36 4.48 RDE 2.76 2.72 3.03 To better reflect anytime convergence behaviour (which dominates the U-score Speed category), we additionally report two speed-focused metrics based on the same median target: (i) the time-to-target (T), defined as the first checkpoint index where a run reaches the target, and (i) an AUC measure computed over 1000 checkpoints as log10(1+max(ft−TGT,0)) _10(1+ (f_t-TGT,0)). Due to the strict 4-page submission limit, detailed per-function T/AUC tables are provided in the supplementary appendix of the full report. In particular, RDEx-SOP shows consistent speed advantages over LSRTDE (T: 17/12/0; AUC: 23/6/0), which helps explain its superior U-score ranking even when final-value differences are often not statistically significant. I-E Algorithm Complexity Let D be the problem dimension, N(g)N^(g) the current front size, and MaxFEMaxFE the evaluation budget. Let TfT_f be the average cost of a single objective evaluation. Per generation, RDEx-SOP performs O(N(g)D)O(N^(g)D) arithmetic operations for mutation/crossover/repair and typically O(N(g)logN(g))O(N^(g) N^(g)) operations for ranking and sampling from the front. The dominant cost is the evaluation cost O(MaxFE⋅Tf)O(MaxFE· T_f), while the algorithmic overhead is approximately O(MaxFE⋅D)O(MaxFE· D) under the standard configuration with N(g)=O(D)N^(g)=O(D). Therefore, for typical CEC benchmarks where evaluation is more expensive than vector operations, the overall runtime is mainly governed by MaxFEMaxFE and the objective-function cost. IV Conclusion This report presented RDEx-SOP and its evaluation on the CEC 2025 SOP benchmark suite. The results demonstrate strong overall U-score performance and competitive final-value statistics across the 29 problems, supporting the effectiveness of combining success-history adaptation with an exploitation-biased hybrid search mechanism. 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] J. Brest, M. S. Maučec, and B. Bošković (2016) iL-SHADE: improved L-SHADE algorithm for single objective real-parameter optimization. In 2016 IEEE Congress on Evolutionary Computation (CEC), p. 1188–1195. Cited by: §I. [2] J. Brest, M. S. Maučec, and B. Bošković (2017) Single objective real-parameter optimization: algorithm jSO. In 2017 IEEE Congress on Evolutionary Computation (CEC), p. 1311–1318. Cited by: §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, 3rd item. [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, 2nd item. [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-E. [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. [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] 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, 1st item, §I-G. [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] 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. [13] 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 Per-function Tables A-A Final-value statistics TABLE IV: Experimental comparison results between RDEx-SOP and other competitors on the 29 benchmark functions in CEC2025 SOP. Problem RDEx-SOP LSRTDE jSOa IEACOP mLSHADE-LR RDE Mean SD Mean SD W Mean SD W Mean SD W Mean SD W Mean SD W 1 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 4.55E-15 6.63E-15 = 2.85E-06 8.17E-07 + 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 2 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 4.32E-14 2.43E-14 + 1.20E-06 1.68E-06 + 5.00E-14 2.45E-14 + 0.00E+00 0.00E+00 = 3 5.90E+01 1.51E+00 5.66E+01 1.08E+01 = 5.86E+01 7.11E-15 - 1.91E+00 1.99E+00 - 6.93E+00 1.49E+01 - 2.18E+01 1.22E+00 - 4 3.82E+00 1.90E+00 3.10E+00 1.39E+00 = 1.07E+01 2.17E+00 + 2.99E+01 7.31E+00 + 8.08E+00 3.07E+00 + 7.72E+00 1.57E+00 + 5 2.36E-07 5.47E-07 5.89E-08 1.70E-07 = 1.23E-13 3.08E-14 = 1.32E-01 2.05E-01 + 2.95E-03 1.03E-02 = 0.00E+00 0.00E+00 = 6 3.63E+01 2.31E+00 3.75E+01 9.42E+00 = 4.07E+01 2.12E+00 + 5.39E+01 5.75E+00 + 3.98E+01 3.11E+00 + 3.87E+01 2.19E+00 + 7 2.39E+00 1.43E+00 2.27E+00 1.75E+00 = 1.11E+01 2.00E+00 + 2.75E+01 6.33E+00 + 7.95E+00 2.61E+00 + 7.40E+00 1.87E+00 + 8 0.00E+00 0.00E+00 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 2.38E+00 9.73E+00 + 0.00E+00 0.00E+00 = 0.00E+00 0.00E+00 = 9 4.61E+02 3.56E+02 3.25E+02 2.20E+02 = 1.40E+03 2.37E+02 + 2.64E+03 4.67E+02 + 1.47E+03 2.86E+02 + 1.41E+03 2.29E+02 + 10 1.63E+00 1.76E+00 1.47E+00 1.87E+00 = 7.38E+00 1.62E+01 + 3.16E+01 1.34E+01 + 8.12E+00 1.08E+01 + 3.20E+00 2.80E+00 = 11 2.14E+00 2.06E+00 1.55E+00 1.71E+00 = 1.74E+02 9.50E+01 + 1.50E+03 4.76E+02 + 1.18E+03 4.25E+02 + 2.34E+02 1.20E+02 + 12 9.43E+00 6.60E+00 1.15E+01 5.59E+00 = 1.95E+01 2.45E+00 + 1.58E+02 1.91E+02 + 1.97E+01 8.35E+00 + 1.52E+01 5.03E+00 + 13 8.17E+00 9.84E+00 1.01E+00 3.95E+00 - 1.24E+01 9.26E+00 + 1.02E+02 2.75E+01 + 2.28E+01 3.35E+00 + 1.89E+01 7.20E+00 + 14 4.88E-01 2.67E-01 4.45E-01 1.67E-01 = 3.27E+00 1.64E+00 + 7.86E+01 8.31E+01 + 1.25E+01 1.29E+01 + 1.36E+00 8.64E-01 + 15 5.58E+00 4.64E+00 4.57E+00 3.37E+00 = 7.83E+01 8.52E+01 + 3.75E+02 1.49E+02 + 5.57E+01 5.71E+01 + 2.69E+01 4.14E+01 + 16 1.56E+01 1.09E+01 2.11E+01 6.36E+00 = 3.36E+01 8.12E+00 + 1.08E+02 6.70E+01 + 3.60E+01 8.99E+00 + 3.01E+01 1.07E+01 + 17 1.17E+01 9.95E+00 1.34E+01 9.67E+00 + 1.84E+01 6.27E+00 + 2.28E+02 5.69E+01 + 3.05E+01 7.41E+00 + 2.08E+01 4.43E-01 + 18 2.02E+00 6.12E-01 2.09E+00 6.08E-01 = 4.81E+00 1.35E+00 + 7.72E+01 7.24E+01 + 1.09E+01 4.78E+00 + 3.11E+00 9.89E-01 + 19 1.32E+01 3.53E+01 1.59E+01 3.50E+01 = 2.99E+01 6.75E+00 + 1.84E+02 3.12E+01 + 4.15E+01 1.02E+01 + 2.54E+01 5.17E+00 + 20 2.02E+02 1.60E+00 2.03E+02 1.75E+00 = 2.11E+02 2.06E+00 + 2.24E+02 6.85E+00 + 2.08E+02 1.98E+00 + 2.07E+02 1.93E+00 + 21 1.00E+02 0.00E+00 1.00E+02 1.67E-13 + 1.00E+02 3.23E-13 - 1.00E+02 1.56E-08 + 1.00E+02 0.00E+00 - 1.00E+02 0.00E+00 - 22 3.40E+02 2.70E+00 3.40E+02 3.70E+00 = 3.54E+02 3.97E+00 + 3.77E+02 1.16E+01 + 3.57E+02 7.35E+00 + 3.47E+02 2.96E+00 + 23 4.17E+02 3.07E+00 4.15E+02 3.93E+00 = 4.28E+02 2.60E+00 + 4.42E+02 3.89E+00 + 4.25E+02 4.35E+00 + 4.23E+02 1.68E+00 + 24 3.87E+02 7.69E-03 3.87E+02 3.66E-03 = 3.87E+02 6.67E-03 = 3.86E+02 1.65E+00 = 3.81E+02 2.62E+00 - 3.79E+02 1.67E-01 - 25 5.89E+02 1.32E+02 6.79E+02 1.14E+02 + 9.43E+02 3.54E+01 + 1.06E+03 4.73E+02 + 9.91E+02 7.41E+01 + 8.99E+02 3.02E+01 + 26 4.70E+02 3.35E+00 4.71E+02 3.57E+00 = 5.00E+02 5.00E+00 + 5.12E+02 6.77E+00 + 5.04E+02 8.88E+00 + 4.71E+02 5.93E+00 = 27 3.00E+02 2.27E-13 3.00E+02 1.83E-13 + 3.09E+02 3.09E+01 = 3.19E+02 4.34E+01 + 3.00E+02 0.00E+00 - 3.09E+02 3.09E+01 - 28 4.02E+02 6.74E+00 4.01E+02 6.37E+00 = 4.31E+02 8.68E+00 + 5.19E+02 4.86E+01 + 4.27E+02 2.02E+01 + 3.66E+02 4.64E+01 - 29 1.98E+03 1.07E+01 1.98E+03 1.35E+01 = 1.97E+03 1.50E+01 - 3.22E+03 5.44E+02 + 1.95E+03 1.65E+02 = 7.66E+02 3.60E+02 - W/T/L −/−/−-/-/- 4/24/1 21/5/3 27/1/1 21/4/4 17/6/6 A-B Speed-side per-function statistics TABLE V: Time-to-target comparison on the 29 CEC2025 SOP functions. The time-to-target is the first checkpoint index (1–1000) where the run reaches the median target (smaller is better); runs that never reach the target are assigned 1001. Problem RDEx-SOP LSRTDE jSOa IEACOP mLSHADE-LR RDE Mean SD Mean SD W Mean SD W Mean SD W Mean SD W Mean SD W 1 538.2 4.1 578.2 3.8 + 785.2 148.1 + 1001.0 0.0 + 805.5 70.6 + 477.0 5.6 - 2 485.9 5.5 514.0 5.0 + 920.7 143.0 + 1001.0 0.0 + 980.5 51.5 + 412.4 4.5 - 3 1001.0 0.0 975.8 123.7 = 1001.0 0.0 = 114.0 41.1 - 811.2 93.9 - 739.9 42.7 - 4 675.7 73.9 705.8 38.3 + 999.0 7.7 + 1001.0 0.0 + 971.5 38.5 + 981.8 21.2 + 5 846.8 136.9 814.3 116.5 = 811.7 65.2 = 1001.0 0.0 + 1001.0 0.0 + 607.2 4.6 - 6 760.0 105.7 874.6 58.7 + 995.4 10.8 + 1001.0 0.0 + 991.4 18.7 + 981.2 19.2 + 7 649.8 38.6 697.0 37.2 + 998.8 6.2 + 1001.0 0.0 + 975.0 48.3 + 985.3 16.5 + 8 454.6 5.0 485.1 3.2 + 567.8 3.1 + 1001.0 0.0 + 431.0 142.2 = 146.8 3.2 - 9 701.2 33.5 728.8 46.2 = 992.7 14.7 + 998.2 13.9 + 970.0 51.6 + 988.2 19.3 + 10 611.5 253.7 556.7 142.8 = 949.2 47.8 + 1001.0 0.0 + 971.1 53.8 + 824.8 132.1 + 11 553.4 49.6 568.0 35.4 = 769.2 193.1 + 1001.0 0.0 + 1001.0 0.0 + 771.3 283.0 = 12 595.2 82.8 674.2 84.9 + 997.4 8.7 + 1001.0 0.0 + 964.4 68.7 + 906.4 95.8 + 13 537.6 74.5 530.9 38.4 = 943.5 52.2 + 1001.0 0.0 + 994.4 25.9 + 944.4 96.3 + 14 540.7 52.1 535.0 30.8 = 984.8 38.5 + 1001.0 0.0 + 995.2 28.4 + 875.6 92.2 + 15 636.8 148.9 618.1 70.4 = 1001.0 0.0 + 1001.0 0.0 + 964.9 53.3 + 942.0 67.3 + 16 706.4 84.9 712.0 42.7 = 996.3 9.1 + 1001.0 0.0 + 986.8 31.2 + 984.1 22.9 + 17 615.2 126.6 781.2 239.4 + 938.6 76.9 + 1001.0 0.0 + 1001.0 0.0 + 911.3 58.8 + 18 620.9 30.1 645.4 39.5 + 997.4 6.7 + 1001.0 0.0 + 1001.0 0.0 + 903.7 64.9 + 19 576.2 129.0 612.4 119.5 + 994.5 11.5 + 1001.0 0.0 + 1000.5 1.8 + 952.6 44.0 + 20 547.3 29.8 595.7 32.5 + 1001.0 0.0 + 1001.0 0.0 + 979.9 35.6 + 973.8 27.6 + 21 689.1 9.8 984.8 38.2 + 704.9 109.4 - 1001.0 0.0 + 321.1 71.8 - 325.3 2.6 - 22 422.2 21.2 444.4 27.5 + 994.5 22.0 + 1001.0 0.0 + 945.9 122.9 + 952.6 36.4 + 23 400.5 24.1 425.5 25.1 + 998.5 12.1 + 1001.0 0.0 + 810.4 257.0 + 955.7 65.2 + 24 832.8 240.3 805.4 234.5 = 955.4 154.8 = 732.6 384.6 = 778.6 92.5 = 132.5 15.8 - 25 334.8 55.6 362.7 19.0 + 994.9 21.0 + 775.7 401.0 + 927.6 200.1 + 787.2 212.8 + 26 293.5 12.8 308.0 20.7 + 1001.0 0.0 + 1001.0 0.0 + 1001.0 0.0 + 392.9 198.5 = 27 906.0 114.3 1001.0 0.0 = 789.3 132.9 - 1001.0 0.0 = 769.3 60.8 - 500.7 152.2 - 28 631.4 82.4 671.6 104.3 + 1001.0 0.0 + 1001.0 0.0 + 983.3 39.7 + 909.6 58.0 + 29 905.8 200.9 817.5 249.1 = 742.2 245.0 - 1001.0 0.0 = 865.8 135.7 = 300.2 59.8 - W/T/L −/−/−-/-/- 17/12/0 23/3/3 25/3/1 23/3/3 18/2/9 TABLE VI: Anytime convergence comparison using AUC over 1000 checkpoints on the 29 CEC2025 SOP 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-SOP LSRTDE jSOa IEACOP mLSHADE-LR RDE Mean SD Mean SD W Mean SD W Mean SD W Mean SD W Mean SD W 1 2.09 0.05 2.34 0.03 + 2.54 0.02 + 0.51 0.02 - 1.98 0.12 - 1.75 0.03 - 2 0.78 0.01 0.82 0.02 + 1.17 0.02 + 1.47 0.29 + 0.71 0.17 = 0.73 0.02 - 3 1.36 0.04 1.32 0.13 = 1.36 0.01 = 0.24 0.08 - 1.35 0.16 = 1.07 0.06 - 4 1.45 0.09 1.53 0.08 + 1.87 0.03 + 1.53 0.12 + 1.98 0.04 + 1.83 0.03 + 5 0.29 0.01 0.30 0.01 + 0.33 0.00 + 0.35 0.12 + 0.14 0.01 - 0.19 0.00 - 6 1.59 0.05 1.92 0.12 + 1.94 0.02 + 1.46 0.12 - 2.03 0.04 + 1.88 0.02 + 7 1.41 0.08 1.51 0.08 + 1.88 0.03 + 1.49 0.09 + 1.96 0.05 + 1.83 0.03 + 8 0.52 0.01 0.56 0.02 + 0.71 0.01 + 1.16 0.54 + 0.29 0.01 - 0.35 0.01 - 9 2.57 0.13 2.67 0.17 = 3.47 0.05 + 3.23 0.10 + 3.36 0.14 + 3.48 0.06 + 10 0.84 0.06 0.89 0.04 + 1.57 0.23 + 1.57 0.17 + 1.60 0.21 + 1.23 0.20 + 11 2.29 0.12 2.38 0.08 + 3.20 0.41 + 3.87 0.16 + 4.24 0.12 + 2.63 0.65 = 12 1.54 0.11 1.71 0.12 + 2.76 0.05 + 2.74 0.38 + 2.57 0.19 + 2.05 0.13 + 13 0.97 0.05 1.00 0.06 = 1.75 0.04 + 2.55 0.30 + 1.75 0.04 + 1.42 0.07 + 14 1.08 0.04 1.13 0.03 + 2.15 0.07 + 2.33 0.35 + 2.14 0.16 + 1.55 0.09 + 15 1.69 0.19 1.70 0.11 = 2.78 0.10 + 2.68 0.17 + 2.71 0.17 + 2.48 0.17 + 16 1.44 0.12 1.50 0.10 = 2.21 0.05 + 1.93 0.32 + 2.19 0.08 + 2.12 0.06 + 17 0.94 0.02 0.99 0.03 + 1.88 0.06 + 3.43 0.32 + 2.02 0.15 + 1.34 0.07 + 18 1.07 0.02 1.13 0.03 + 2.16 0.03 + 2.13 0.23 + 2.08 0.08 + 1.59 0.05 + 19 1.29 0.32 1.37 0.31 + 2.32 0.05 + 2.34 0.09 + 2.29 0.08 + 2.15 0.07 + 20 1.17 0.07 1.27 0.07 + 1.86 0.03 + 1.42 0.12 + 1.94 0.04 + 1.78 0.04 + 21 0.42 0.01 0.48 0.01 + 0.55 0.01 + 0.14 0.01 - 0.30 0.01 - 0.35 0.01 - 22 0.91 0.04 0.96 0.05 + 1.89 0.04 + 1.65 0.12 + 1.78 0.25 + 1.78 0.06 + 23 0.86 0.05 0.91 0.05 + 1.84 0.04 + 1.47 0.08 + 1.22 0.28 + 1.73 0.09 + 24 0.32 0.01 0.35 0.01 + 0.36 0.01 + 0.24 0.08 - 0.60 0.08 + 0.23 0.01 - 25 1.07 0.18 1.15 0.05 + 2.84 0.07 + 2.17 1.12 + 2.35 0.47 + 2.09 0.38 + 26 0.53 0.02 0.55 0.03 + 1.38 0.11 + 1.49 0.10 + 1.55 0.09 + 0.51 0.20 = 27 0.52 0.06 0.54 0.03 + 0.94 0.37 + 0.58 0.67 - 1.48 0.19 + 0.80 0.40 + 28 1.45 0.13 1.53 0.12 + 2.39 0.04 + 2.25 0.12 + 2.39 0.09 + 2.18 0.10 + 29 2.01 0.29 1.96 0.35 = 2.10 0.25 = 3.51 0.19 + 3.01 0.34 + 1.22 0.10 - W/T/L −/−/−-/-/- 23/6/0 27/2/0 23/0/6 23/2/4 19/2/8