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

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.

• The paper introduces RDEx-MOP, a fixed-budget multiobjective differential-evolution solver combining IBEA selection, niche preservation, adaptive elite shrinking, and complementary exploration operators.
• RDEx-MOP achieved the highest CEC 2025 total U-score—36,343.5—and best average rank of 1.40, reaching median targets in roughly 400–1,800 evaluations across benchmark problems.
• The method decisively outperformed three non-RDEx competitors, but its advantage over the RDEx baseline was small and not statistically significant after Holm correction, highlighting the need for component ablation and broader testing.

RDEx-MOP is a differential evolution (DE) solver designed for the bound-constrained multiobjective track of the IEEE CEC 2025 numerical optimisation competition, where performance is scored by the median-target U-score framework rather than by final IGD alone. The paper documents the algorithm's design, its reconstruction of an earlier two-stage competition backbone, and its official benchmark results, in which it attains the highest total U-score and best average rank among all released comparison algorithms (2603.27092).

Motivation and positioning

The CEC 2025 MOP track evaluates solvers on how quickly they reach target regions under a fixed budget of 100,000 function evaluations, in addition to final IGD quality. This shifts the design objective from producing a good final front to achieving rapid early IGD descent while preserving diversity. RDEx-MOP is positioned at the intersection of two research lines: the classical evolutionary multiobjective selection paradigms (NSGA-II's non-dominated sorting, SPEA2's strength-based fitness, IBEA's indicator-based comparison, MOEA/D's decomposition), and the single-objective DE lineage from JADE through SHADE and L-SHADE, which established that top-ranked current-to-$p$best variation with shrinking elite windows improves fixed-budget search. The authors explicitly state that these single-objective mechanisms should not be transplanted mechanically into multiobjective settings; only components that strengthen early IGD descent are retained, while archive maintenance is delegated to the multiobjective selector. The algorithm also builds directly on TEMOF, a two-stage framework that improved fixed-budget IGD behaviour on the 2024 competition suite.

Algorithmic design

RDEx-MOP maintains three structures: a working population $P$, an auxiliary population $P_2$ activated after half the budget via a random switching rule, and a Pareto-candidate set $PC$ extracted from nondominated solutions with niche-based maintenance. Environmental selection follows IBEA: objectives are normalised component-wise, an additive $\epsilon$-style indicator matrix is computed, and fitness is assigned via exponential discounting with indicator parameter $\kappa = 0.05$. Worst individuals are removed iteratively with incremental fitness updates, providing convergence pressure without crowding-distance sorting.

Diversity is handled by two complementary mechanisms. First, when $PC$ exceeds the population size, a niche maintenance procedure estimates a radius from mean third-nearest-neighbour distances and deletes solutions with the largest multiplicative crowding score $1 - \prod_j R_{ij}$. Second, a niche exploration operator identifies $PC$ members in sparse niches (at most one neighbour within threshold radius) and applies a lightweight "half-DE" mutation $v = x + F(x_{r_1} - x_g)$ with Cauchy-sampled $P$0 and normally sampled crossover rate.

Exploitation uses DE/current-to-$P$1best/1 with a discrete parameter pool ($P$2, $P$3), a shrinking elite window from $P$4 down to 2 as evaluations progress, and post-crossover Cauchy perturbation applied per-component with probability 0.2. Computational complexity is $P$5 per generation for both the indicator matrix and the distance-based niche operators, which is standard for indicator-based selection at $P$6.

Benchmark results

The evaluation covers MaOP1–MaOP10 with 30 runs each, IGD recorded every 200 evaluations across 500 checkpoints. Against the released competitors (RDEx, TEMOFNSGA3, TFBCEIBEA, TGFMMOEA), RDEx-MOP achieves a total U-score of 36343.5 versus 35956.5 for the RDEx baseline, with average rank 1.40 versus 1.60. Both RDEx variants reach the median target on all runs, so their scores are dominated entirely by the Speed category — a notable structural feature of the metric, since Accuracy contributions drop to zero once targets are always met.

The supplementary diagnostics sharpen this picture considerably:

• Against non-RDEx competitors, the separation is categorical: RDEx-MOP wins all 10 functions on Final IGD, TTT, and AUC against TEMOFNSGA3, TFBCEIBEA, and TGFMMOEA, with Holm-corrected significance throughout and Vargha–Delaney $P$7 effect sizes of 1.00.
• Against the RDEx baseline, differences are marginal: uncorrected Wilcoxon gives 2/8/0 on Final IGD and TTT, but Holm correction reduces this to 0/10/0 on both metrics, and $P$8 values of 0.55 and 0.52 indicate near-negligible effects. On AUC, RDEx actually leads slightly ($P$9, one loss for RDEx-MOP).
• Friedman tests over per-function medians confirm overall ordering (Final IGD: $P_2$0, $P_2$1), with RDEx-MOP ranked first on Final IGD (1.40) and tied with RDEx on TTT (1.50 each), while RDEx holds the best AUC rank (1.20 vs 1.80).

The time-to-target data illustrate the speed gap concretely: RDEx-MOP reaches median targets within roughly 2–9 checkpoints (i.e., 400–1,800 evaluations out of 100,000), whereas the three non-RDEx competitors require hundreds of checkpoints or fail to reach targets at all. This means the practical contribution of RDEx-MOP over RDEx is a small but consistent improvement concentrated in problems 3, 8, 9, and 10, rather than a qualitative advance.

Limitations and open questions

Several caveats bear directly on the reported results. First, the margin over the RDEx baseline is not statistically significant after Holm correction on any of the three diagnostic metrics; the first-place ranking rests on small per-function advantages accumulated under the U-score aggregation. Second, because both RDEx variants hit the median target on every run, the official metric cannot distinguish them beyond checkpoint-level timing granularity, leaving final-IGD quality as the only differentiator — where the advantage is likewise not Holm-significant. Third, the evaluation is confined to the ten CEC 2025 MOP problems with a fixed population size of $P_2$2 and the specific discrete $P_2$3 pool; no ablation isolates the contribution of individual components (niche exploration, Cauchy perturbation, the auxiliary population switching rule), so the source of the improvement over RDEx remains unidentified. Fourth, the paper does not report sensitivity to $P_2$4 or to the niche-radius estimator based on third-nearest-neighbour distances. Whether the framework generalises to many-objective settings ($P_2$5), constrained problems, or larger budgets is left open.

Conclusion

RDEx-MOP combines IBEA-style indicator selection, niche-maintained Pareto-candidate archives, and complementary DE operators tuned for rapid early convergence, achieving first place in the official CEC 2025 MOP U-score evaluation. The evidence supports a clear dominance over the non-DE competitors and a statistically indistinguishable but consistently favourable position relative to its RDEx predecessor. The main open question raised by the results is which of the reconstructed components accounts for the residual gains, given that no component-wise ablation is provided.