Bayesian optimization (BO) is effective for expensive black-box optimization, but its performance often degrades on multimodal or high-dimensional problems. A promising remedy is to run multiple local search processes, such as trust region--based BO, and allocate evaluations to those with high potential. However, current best values do not always reflect the final performance of each process. We propose a trajectory-aware best-arm identification (BAI) framework that uses optimization history to guide this allocation. The proposed method extrapolates improvement trajectories, estimates final performance, and progressively eliminates low-potential candidates. For trust region--based BO, we theoretically show that the proposed BAI-guided allocation accelerates convergence to the global optimum under mild assumptions. Experiments on synthetic and real-world benchmarks demonstrate that our method improves BO on multimodal problems. We also show that the same mechanism can enhance other local or population-based optimizers, suggesting its potential as a general extension strategy for multimodal black-box optimization.

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