Process-Aware LNS for Large-Scale MILP via Context-Enhanced Fine-Tuned LLM-driven Selector

Solving large-scale Mixed-Integer Linear Programming (MILP) problems involving millions of variables is critical for industrial applications but notoriously intractable due to combinatorial explosion. While Large Neighborhood Search (LNS) has emerged as the premier heuristic strategy for tackling such scales, its efficacy hinges entirely on the underlying neighborhood selection mechanism. Recent Large Language Model (LLM)-driven LNS frameworks show remarkable promise but suffer from fundamental flaws: they lack generalizability across diverse problem classes and rely on static selection paradigms that remain entirely blind to the shifting dynamics of the iterative optimization process. To overcome this rigidity, we propose PALL, a Process-Aware Evolutionary LNS framework guided by a Context-Enhanced Fine-Tuned LLM-driven Selector. Instead of deploying a monolithic operator, PALL leverages an offline-evolved diverse operator ensemble and dynamically synchronizes the search strategy with evolving optimization states. Specifically, we fine-tune an 8B-parameter LLM on high-quality hindsight oracle trajectories to adaptively perform optimal operator selection. To bolster the LLM's sequential decision-making, we introduce a context-enhanced reasoning mechanism that leverages historical operator trajectories and optimization rewards as inferential feedback. Extensive evaluations on standard million-variable MILP benchmarks demonstrate that PALL achieves state-of-the-art performance. Supported by an asynchronous deployment strategy, it not only consistently outperforms commercial solvers like Gurobi and classical LNS algorithms but also delivers over a $6\times$ acceleration compared to state-of-the-art LLM-driven LNS frameworks.