Although Multi-Vector Retrieval (MVR) has achieved the state of the art on many information retrieval (IR) tasks, its performance highly depends on how to decompose queries into smaller pieces, say phrases or tokens. However, optimizing query decomposition for MVR performance is not end-to-end differentiable. Even worse, jointly solving this problem and training the downstream retrieval-based systems, say RAG systems could be highly inefficient. To overcome these challenges, we propose Performance-Oriented Query Decomposer (POQD), a novel query decomposition framework for MVR. POQD leverages one LLM for query decomposition and searches the optimal prompt with an LLM-based optimizer. We further propose an end-to-end training algorithm to alternatively optimize the prompt for query decomposition and the downstream models. This algorithm can achieve superior MVR performance at a reasonable training cost as our theoretical analysis suggests. POQD can be integrated seamlessly into arbitrary retrieval-based systems such as Retrieval-Augmented Generation (RAG) systems. Extensive empirical studies on representative RAG-based QA tasks show that POQD outperforms existing query decomposition strategies in both retrieval performance and end-to-end QA accuracy. POQD is available at https://github.com/PKU-SDS-lab/POQD-ICML25.