Understanding Large Language Models as Numerical Optimizers

The paper examines large language models (LLMs) as tools for numerical optimization, focusing on their ability to exploit mathematical problem structures and generate near‑optimal solutions. An evaluation framework is introduced to separate solution quality from execution failures such as numerical inaccuracies, constraint violations, and formatting errors. Experiments on convex optimization, fractional programming, and constrained projection problems show that LLMs can capture useful optimization structures, but reliability decreases with higher dimensionality and depends on the problem’s structure. An iterative refinement mechanism that provides targeted feedback on feasibility and solution quality significantly improves reliability and extends the dimensional range where LLMs produce valid solutions, though persistent numerical and structural errors limit their use as standalone optimizers.

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