DE-2LS: Differential Evolution with Late-Stage local-search for Unconstrained Single-Objective Numerical Optimization

Technical Summary: DE-2LS for Unconstrained Single-Objective Numerical Optimization

Problem Statement
The paper addresses unconstrained single-objective numerical optimization problems (UCOPs) with variable bounds, a setting common in IEEE CEC competitions. The core challenge lies in balancing global exploration, late-stage exploitation, and function-evaluation efficiency. Specifically, while Differential Evolution (DE) variants like RDEx (Reconstructed DE with exploitation-biased branching) demonstrate strong performance, they may lack the precision to efficiently "polish" incumbent solutions once the search enters a promising basin. This limitation is critical under the U-score evaluation metric, which jointly assesses final solution quality and convergence speed. The goal is to improve both metrics without disrupting the robust evolutionary search behavior of the baseline algorithm.

Methodology
The proposed method, DE-2LS, is a conservative enhancement of the RDEx framework. Rather than replacing RDEx's core evolutionary engine, DE-2LS introduces two late-stage refinements designed to operate within a fixed function-evaluation budget:

1. Smoothed Exploitation-Biased (EB) Branch-Rate Update:
   In the late stage of the search (after 75% of the evaluation budget is consumed), the mechanism updates the probability of selecting the exploitation-biased branch. Instead of reacting abruptly to immediate branch performance, the update rule applies a weighted smoothing (linear combination of the previous rate and the current raw estimate). This stabilizes the transition to intensive exploitation, preventing premature convergence or erratic shifts in search strategy.

2. Guarded Coordinate-Pattern Local Search (LS):
   A lightweight local-search module is activated only when specific conditions are met: the search is in the late stage (after 82% of the budget), the global-best solution has stagnated for a defined number of generations, and the local-search call limit has not been reached.

   • Mechanism: The LS operates on the current global-best solution using a coordinate-pattern approach. It explores one coordinate direction at a time with an initial step size proportional to the variable bounds.
   • Strategy: It employs a "first-improvement" strategy; if a better point is found, the center is updated immediately. If no improvement is found, the step size is halved.
   • Constraints: The process is strictly budget-aware, with a defined ratio of evaluations allocated per call and a maximum number of calls. It uses objective-only acceptance (since the problem is unconstrained except for bounds) and reinjects improved solutions into the population without resetting the success-history memory or the population structure.

Key Contributions

• Algorithm Design: The proposal of DE-2LS, which augments the RDEx baseline with a smoothed EB-rate update and a guarded, budget-controlled local-search module.
• Ablation Study: A rigorous, staged ablation analysis was conducted to determine the optimal configuration. This included tuning the timing and weights of the EB-rate smoothing, the initial EB-rate, the Gaussian sampling scale for the standard branch, selection-pressure parameters, and the local-search coefficients (timing, budget, and step size).
• U-Score Optimization: The design explicitly targets the U-score metric, aiming to improve both the accuracy (final error) and speed (convergence trajectory) components simultaneously.

Experimental Results
Experiments were conducted on 29 UCOP functions (D=30) with a budget of 10,000D function evaluations, using 25 independent runs per algorithm. Performance was evaluated using the pairwise U-score procedure.

• Comparison with RDEx: DE-2LS consistently outperformed the original RDEx. The U-score increased from 33,602.0 to 37,448.0, representing an 11.45% improvement. This gain was driven by a 17.25% improvement in the speed component and a 5.95% improvement in the accuracy component.
• Comparison with Competitive Algorithms: In a broader comparison against several competitive and IEEE CEC-winning algorithms (including LSRTDE, BlockEA, mLSHADE-RL, jSO, and IEACOP), DE-2LS achieved the highest overall U-score of 178,966.5. This result outperformed the second-best algorithm (RDEx) by 3.17% and the average of the other competitors by approximately 34.43%.
• Function-Wise Performance: DE-2LS achieved the best U-score on 11 out of 29 functions, the highest count among all compared algorithms. It demonstrated a particular strength in convergence speed, reaching competitive error levels earlier than rivals on many functions.

Significance and Claims
The paper claims that DE-2LS demonstrates that a carefully designed late-stage local-search strategy can significantly enhance the performance of a strong baseline DE algorithm without compromising its global search capabilities. The authors emphasize that the "do no harm" design principle—preserving the original RDEx engine while adding a lightweight, budget-aware polishing mechanism—is effective for improving both convergence speed and final objective quality. The results suggest that the proposed smoothed EB-rate control and guarded local search provide a robust balance between exploration and exploitation, particularly under the U-score evaluation framework which penalizes slow convergence. The source code for DE-2LS is made available to support reproducibility.