A New Explicit Penalty Method for Evolutionary Multimodal Optimization
Bibliographic record
Abstract
When employing evolutionary algorithms (EAs) to solve multimodal optimization problems (MMOPs), effectively utilizing diversity information is crucial to prevent the population from converging to a single peak. This requires balancing diversity and objective value—a challenge that inherently constitutes a penalty problem. Although some implicit penalty methods have been proposed to address this issue, most lack flexibility in penalty formulation. In this study, we present a novel explicit penalty method (EPM) designed to effectively leverage diversity information for multimodal optimization. First, the diversity of a solution is quantified by its distance to the nearest neighbor with a better objective value. Then, an explicit penalty function is formulated by integrating diversity and objective value. This function facilitates the capture of multiple peaks and balances the search among them. If a reasonable number of peaks are identified, a local search is applied to each for refinement; otherwise, a global search is conducted across the decision space. Through this adaptive process, EPM locates multiple optima both efficiently and accurately. Extensive experiments demonstrate that EPM outperforms several multimodal optimization methods, including 11 popular approaches, eight recent state-of-the-art algorithms, and an IEEE CEC competition winner. Moreover, even when integrated with classic differential evolution (DE), EPM exhibits highly competitive performance.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".