A survey on mathematical reasoning and optimization with large language models

Mathematical reasoning and optimization are central to artificial intelligence (AI) and computational problem-solving. Recent advancements in Large Language Models (LLMs) have greatly improved AI-driven mathematical reasoning, theorem proving, and optimization techniques. This survey reviews the evolution of mathematical problem-solving in AI, from early statistical learning to deep learning and transformer-based approaches. We examine the capabilities of pre-trained models and LLMs in handling arithmetic operations, complex reasoning, theorem proving, and symbolic computation. A major focus is how LLMs integrate with optimization and control frameworks, including mixed-integer programming, linear-quadratic control, and multi-agent optimization. LLMs assist in problem formulation, constraint generation, and heuristic search, bridging theoretical reasoning with practical applications. We analyze enhancement techniques such as Chain-of-Thought reasoning, instruction tuning, and tool-augmented approaches that strengthen LLM problem-solving. Despite progress, LLMs still face challenges in numerical precision, logical consistency, and proof verification. Emerging solutions include hybrid neural-symbolic reasoning, structured prompt engineering, and multi-step self-correction to enhance reliability. Future research should prioritize interpretability, integration with domain-specific solvers, and robustness in decision-making. This survey provides a comprehensive overview of current capabilities and future directions for mathematical reasoning and optimization with LLMs, highlighting applications in engineering, finance, and scientific research.