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

Abstract: Constrained multiobjective optimisation requires fast feasibility attainment together with stable convergence and diversity preservation under strict evaluation budgets. This report documents RDEx-CMOP, the differential evolution variant used in the IEEE CEC 2025 numerical optimisation competition (C06 special session) constrained multiobjective track. RDEx-CMOP integrates an ε-level feasibility schedule, a SPEA2-style indicator-driven fitness assignment, and a fitness-oriented current-to-pbest/1 mutation operator. We evaluate RDEx-CMOP on the official CEC 2025 CMOP benchmark using the median-target U-score framework and the released trace data. Experimental results show that RDEx-CMOP achieves the highest total score and the best overall average rank among all released comparison algorithms, with strong target-attainment behaviour and near-zero final violation on most problems.

• The paper presents RDEx-CMOP, which uniquely combines an ε-feasibility schedule, SPEA2-style fitness, and indicator-guided DE operators to accelerate feasible Pareto front discovery.
• It achieves top performance on the CEC 2025 CMOP benchmarks by consistently outperforming competitors in U-score, speed, and accuracy under strict evaluation budgets.
• The study demonstrates that coupling adaptive DE variation with constraint handling is critical for rapid and reliable convergence in fixed-budget constrained multiobjective optimization.

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

Problem Statement and Context

Constrained multiobjective optimization problems (CMOPs) are characterized by the simultaneous requirements of driving the population toward feasibility, preserving diversity, and achieving convergence under limited function evaluation (FE) budgets. The Pareto-optimal front often lies in a highly restricted feasible region, making efficient feasibility attainment and rapid front approximation under fixed FE constraints central for algorithmic competitiveness, particularly in benchmarks such as the CEC 2025 CMOP track. Classical approaches—NSGA-II, SPEA2, IBEA, and MOEA/D—have established the importance of combining convergence-diversity balancing with constraint handling mechanisms, often via feasibility rules or $\varepsilon$-level control.

Recent advances indicate that strong performance in fixed-budget CMOP tracks requires tight integration between feasibility management, indicator-guided fitness assignment, and exploitation-biased variation. Differential Evolution (DE) remains a central paradigm, especially when enhanced by JADE-style current-to-$p$best operators and adaptive variation schedules. This paper introduces RDEx-CMOP, a DE-based CMOP solver constructed for the CEC 2025 competition, emphasizing feasibility-aware selection, SPEA2-style fitness, and fitness-oriented search bias.

Algorithmic Framework

Feasibility-Guided Search and Selection

RDEx-CMOP leverages an $\varepsilon$-scheduled constraint-handling mechanism. The initial population's maximum constraint violation $\varepsilon_0$ decays over time to zero, following a schedule designed to fully suppress constraint violations by the midpoint of the FE budget. Solutions with $\mathrm{CV}(x)\le\varepsilon(\mathrm{FE})$ are eligible for selection, focusing the search on rapid feasible region discovery. The SPEA2-style strength-density fitness function is modified to operate within this $\varepsilon$-feasible context and orchestrates selection and truncation, prioritizing diversity while always enforcing feasibility dominance.

Indicator-Guided DE Backbone

The search mechanism is a variation of the DESDE-style DE backbone with several salient innovations:

• Current-to-$p$best/1 Mutation: The mutation follows a fitness-oriented selection, adaptively shrinking the $p$-best parent window over the evolutionary process to concentrate exploitation as the search progresses.
• Parameter Pool: The discrete mutation and crossover pools $F\in\{0.6, 0.8, 1.0\}$ and $CR\in\{0.1, 0.2, 1.0\}$ enhance local search adaptability.
• Directional Differential Term: The sign of the scaling factor in differential variation is determined based on the relative fitness of randomly selected parents, adding a fitness-driven search bias.
• Cauchy Local Perturbation: A lightweight local search is stochastically applied to non-recombined variables, integrating exploratory capabilities without excessive computational overhead.

These components collectively yield an $p$0 per-generation cost, with computational bottlenecks dominated by the strength-density calculation and environmental selection.

Evaluation Protocol

All results reported use the official CEC 2025 SDC1–SDC15 CMOP test suite, with 30 independent runs per problem and a strict budget of 200,000 FEs. The performance metric of record is the U-score, emphasizing both target attainment speed and final IGD. Results are disseminated using the official trace and target-attainment-based evaluation package, ensuring direct comparability with all other released competitive algorithms.

Empirical Results and Analysis

Benchmark Outcomes

RDEx-CMOP outperforms five competitive algorithms, notably DESDE, CCEMT, CCPTEA, IMTCMO, and MTCMMO. Key numerical achievements include:

• Highest total U-score: RDEx-CMOP attains $p$1, with a leading average per-problem score and a top average rank ($p$2).
• Statistically consistent wins: According to the official median target metric, RDEx-CMOP leads in Speed and maintains strong Accuracy and Constraint scores, consolidating its advantage on the majority of benchmark functions.
• Wilcoxon and Friedman Significance: Pairwise Wilcoxon tests show $p$3-$p$4 wins for robust final quality ($p$5) and $p$6-$p$7 function wins on time-to-target (TTT), with median Vargha–Delaney $p$8 effect size significantly above $p$9 for all major competitors. Friedman tests are highly significant for both $\varepsilon$0 ($\varepsilon$1) and TTT ($\varepsilon$2).

Constraint-Driven Performance

The final violation diagnostics consistently evidence near-zero average violations on most problems for RDEx-CMOP, supporting the efficacy of the $\varepsilon$3-schedule. On the split-IGD and constraint-violation axes (as complementary diagnostics), the method is either dominant or statistically indistinguishable from the best competitor.

Computational Considerations

The $\varepsilon$4 strength-density fitness remains a nontrivial cost at large $\varepsilon$5 or $\varepsilon$6, but in practice, the FE-dominated wall-clock time means this is not a bottleneck for problems in the CEC 2025 suite.

Theoretical and Practical Implications

RDEx-CMOP demonstrates that explicit $\varepsilon$7-feasibility schedules, if tightly coupled with indicator-based selection and fitness-oriented variation, yield measurable improvements in feasible target attainment and final front approximation under evaluation constraints. By restricting the search to the most relevant DE advances—decoupled from overcomplicated or over-general multioperator/parameter adaptation regimes—and focusing on efficient feasibility guidance, RDEx-CMOP provides evidence for minimal but critical design transfer from recent DE and CMOP competition backbones.

This work confirms that, in fixed-budget CMOP tracks where interim progress is central, joint design of constraint management and selection/variation algorithms is critical. Further, the algorithmic choices in RDEx-CMOP imply that continued emphasis on task-relevant indicator integration and adaptive feasibility control will remain central to competitive CMOP solving.

Future Directions

Potential avenues for future research include:

• Generalizing the $\varepsilon$8-schedule and SPEA2-style selection to multi-task or transfer optimization for highly dynamic constraint environments.
• Investigating the interaction between parameter pool discretization and continuous adaptation in high-dimensional problems or under severely multimodal constraints.
• Applying the indicator-guided DE paradigm to real-world engineering CMOPs and hybrid approaches, including surrogate-assisted evolutionary systems.

Conclusion

RDEx-CMOP delivers state-of-the-art fixed-budget constrained multiobjective search by merging an $\varepsilon$9-feasibility schedule, indicator-guided SPEA2 selection, and fitness-driven DE exploitation. The method secures leading performance on the CEC 2025 CMOP suite, establishing a new reference for efficient, feasibility-aware Pareto optimization under strict evaluation budgets (2604.03708).