CMA-ES is, per run, a local optimizer; multimodal search relies on restart strategies such as IPOP and BIPOP, which draw every restart uniformly and reuse no information from previous evaluations. Multi-Start Clustering CMA-ES (MSC-CMA-ES) makes restarts structure-aware: in alternating cycles, a Sobol pre-sample is partitioned into approximate basins of attraction by nearest-better clustering, restarts are seeded basin by basin with locally scaled step sizes and population sizes, redundant basin visits are detected and excluded, and the remaining budget is spent on an unbounded local refinement of the best-so-far solution. We evaluate the method on four CEC suites (CEC2014, CEC2017, CEC2020, CEC2022) at their official budgets, across ten (suite, dimension) cells with dimensions 5-30, 51 runs per function, against BIPOP-CMA-ES and five differential-evolution algorithms (ARRDE, jSO, j2020, NL-SHADE-RSP, LSRTDE). Read per function class, MSC-CMA-ES leads on one class, is mixed on a second, and trails on the third. On composition functions, MSC-CMA-ES attains the best value on all four aggregate measures, with 2.7x the fixed-budget target coverage of BIPOP-CMA-ES - the highest composition coverage of any algorithm evaluated. On basic functions, it achieves the best (lowest) median error but exhibits a lower deep-target coverage - the measured price of spending budget on landscape discovery. On hybrid functions both CMA variants trail the leading DE algorithms; the deficit belongs to the CMA family, not to the restart mechanism. All results and scripts are publicly available.

翻译：CMA-ES每次运行均为局部优化器；多模态搜索依赖IPOP和BIPOP等重启策略，这些策略每次均匀采样重启点，且不复用先前评估中的信息。多起点聚类CMA-ES（MSC-CMA-ES）使重启具有结构感知能力：在交替循环中，通过最近优聚类将Sobol预采样点划分为近似吸引盆地，逐盆地以局部尺度缩放步长和种群规模进行重启播种，检测并排除冗余盆地访问，剩余预算用于对当前最优解进行无界局部精化。我们在四个CEC测试套件（CEC2014、CEC2017、CEC2020、CEC2022）的官方预算下，对维度5-30的十个（套件，维度）组合，每个函数运行51次，与BIPOP-CMA-ES及五种差分进化算法（ARRDE、jSO、j2020、NL-SHADE-RSP、LSRTDE）进行了对比。按函数类别分，MSC-CMA-ES在一类上领先，一类上表现混合，一类上落后。在组合函数上，MSC-CMA-ES在全部四项聚合指标上取得最优值，固定预算目标覆盖率为BIPOP-CMA-ES的2.7倍——在所有评估算法中组合覆盖率最高。在基础函数上，其取得最优（最低）中位数误差，但深度目标覆盖率较低——这是将预算用于景观发现所付出的代价。在混合函数上，两种CMA变体均落后于领先的DE算法；该差距属于CMA算法家族本身，而非重启机制所致。所有结果与脚本均已公开。