Large Language Models (LLMs) have significantly advanced the automation of scientific discovery. However, existing systems share a core limitation: they optimize ideas independently for either Quality or Diversity. This often results in closely related ideas or large sets of trivial, unsound, or unclear concepts. We instead argue that research ideation should be framed as a conjunction of Quality and Diversity objectives, i.e., a Quality-Diversity (QD) search. Accordingly, we introduce IDEAgent, a multi-agent framework that tracks idea evolution through lineages. We jointly drive Quality using multi-objective feedback for repair and refinement, while Diversity is achieved through lightweight sequential memory and explicit comparison against completed ideas, historical variants, and rejected proposals. To evaluate this joint objective, we develop Yield, a metric that computes the largest mutually diverse set of ideas satisfying predefined quality thresholds. Across 32 topics spanning 8 CS domains, IDEAgent outperforms the best baseline by 3.89x on Yield, while achieving Yield of at least 2 on 8x as many topics. We further analyze quality improvements, showing that repair and refinement improve logical rigor and clarity while preserving non-obviousness. Lastly, we extend IDEAgent to two algorithm discovery tasks for verifiable quality judgment and find that it still consistently outperforms all baselines, and yields the largest set of high-quality and diverse programs that beat a trivial/standard baseline. To encourage future research on QD-search-based ideation, we open-source IDEAgent at https://github.com/declare-lab/IDEAgent.

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