Design spaces serve as a conceptual framework that enables designers to explore feasible solutions, yet their lack of computational formalization limits integration with AI-driven design automation. To address this, we introduce a structured design space model that formalizes design spaces with orthogonal dimensions and discrete elements, making them machine-interpretable and executable. Building on this model, we present IDEA, a fully automated design exploration framework to generate effective outcomes based on user requirements and design spaces. Specifically, IDEA leverages large language models (LLMs) for constraint generation, incorporates a constraint-guided Monte Carlo Tree Search (MCTS) algorithm to explore the space, and instantiates abstract decisions into domain-specific implementations. We evaluate IDEA in two design scenarios: data-driven article and standard visualization, supported by comparativeratings, expert interviews, and quantitative experiments. Results demonstrate the IDEA's adaptability across domains and its capability to produce high-quality designs.