Chaotic evolution with online region discovery for multimodal optimization

Multimodal optimization aims to locate and maintain multiple high-quality solutions in complex, high-dimensional landscapes. Classical niching-based evolutionary algorithms mainly operate at the level of individual distances or densities, and typically maintain diversity in the objective space without explicitly reconstructing the underlying peak–basin structure in the decision space. As a result, they are prone to pseudo-multimodality: a large number of seemingly different solutions may in fact concentrate around only a few basins of attraction. To address this issue, we propose Chaotic Landscape-Decoding Evolution (CLDE), a decision space-centric evolutionary framework that couples chaotic evolution exploration with persistence-guided basin decoding and basin-wise refinement. CLDE first drives each individual by a logistic chaotic map with a decaying step-size schedule, generating structurally rich trajectories that cross basin boundaries in early stages and gradually concentrate around promising regions. On the pooled candidates we build a k-nearest-neighbor graph and define a unified height function that uses the objective value in single-objective problems and a rank–crowding indicator in multi-objective problems. A persistence-guided basin growing procedure is then performed on this graph to merge peaks that are not separated by sufficiently deep valleys, yielding a compact set of topologically stable basins. To reduce sensitivity to a fixed merging scale, we further introduce an adaptive persistence threshold that adjusts the decoding resolution online according to the evolving sample distribution, mitigating both over-fragmentation and over-merging without costly parameter sweeping. Within each basin, CLDE employs a dual-factor selection score that combines objective quality and structural representativeness, and applies local Gaussian refinement to jointly improve solution quality and coverage. We instantiate the framework as CLDE-S for single-objective multimodal optimization and CLDE-M for multi-objective multimodal optimization. On 20 CEC2013 single-objective multimodal benchmarks, CLDE-S achieves the highest average peak ratio among representative multimodal evolutionary algorithms and attains perfect peak coverage on most low- and medium-dimensional functions. On DTLZ and MMMOP test suites, CLDE-M achieves competitive IGD and decision-space coverage compared with representative multi-objective and multimodal multi-objective optimizers. These results indicate that explicit landscape decoding in the decision space provides a robust and scalable basis for multimodal evolutionary search in both single- and multi-objective settings.