OptMaster: A DAG-Based Framework for Formulation and Heuristic Discovery in Optimization

Abstract

Optimization problems are fundamental across science and industry, including planning, scheduling, and resource allocation. While LLMs show promise in automating optimization, they struggle to bridge the gap between real-world requirements and both mathematical formulations and effective heuristic designs. Furthermore, the field lacks a unified framework that spans problem formulation and heuristic discovery for NP-hard settings. To address these challenges, we propose OptMaster, a unified framework that spans optimization from formulation to heuristic discovery, structuring the process as a Directed Acyclic Graph (DAG) where each node represents a candidate solution. The DAG architecture enables cross-branch knowledge transfer when search progress stagnates. Within each node, we further replace textual self-reflection with independently generated verification code, grounding the evaluation in deterministic computation to suppress hallucinations. OptMaster achieves competitive performance across two optimization paradigms. In Formulation Intelligence, OptMaster achieves state-of-the-art accuracy across the three most challenging benchmarks in the field. In Heuristic Discovery, OptMaster surpasses the best known solutions on Circle Packing ($n=26, 32$) and achieves a cut of 9,590 on Gset70 with significantly reduced time and search budgets.