From Heuristic Selection to Automated Algorithm Design: LLMs Benefit from Strong Priors

Large language models (LLMs) are increasingly used for optimization modeling and solver-code generation, yet practical operations research and optimization problems often require a harder capability: designing scalable algorithms that exploit problem structure and outperform direct formulation-and-solve baselines. Existing benchmarks are limited to small or simplified examples far below real-world scale and complexity. We introduce FrontierOR, among the first benchmarks to systematically evaluate LLM-based efficient algorithm design for realistic large-scale optimization problems. FrontierOR includes 180 tasks derived from methodologically diverse papers published in top-tier operations research venues, each with standardized instances and a hidden, expert-verified evaluation suite. We evaluate seven LLMs spanning frontier, cost-effective, and open-source models both in one-shot and test-time evolution settings. The results reveal that frontier models still struggle to move from executable formulations to efficient optimization algorithms: the strongest one-shot model outperforms Gurobi in only 31% of cases in both solution quality and computational efficiency, and even strong coding agents with test-time evolution achieve only 50% on selected hard tasks. FrontierOR establishes a practical evaluation platform for LLM-based optimization algorithm design, which enables future LLMs and agents to be systematically tested on whether they can move beyond correct formulation toward a feasible, high-quality, and efficient algorithm. Code and data are publicly released at https://github.com/Minw913/FrontierOR.

LLMs now prove hard theorems in Lean. Can they also design algorithms and prove their correctness and efficiency? LLM-guided design of optimisation heuristics (such as AlphaEvolve) is purely empirical. Automated theorem provers with LLMs prove given statements. So far, there has been little evidence that LLMs can do both: design an algorithm and guarantee its efficiency by a mathematical proof. We present a staged pipeline modelled after research methodology in design and analysis of evolutionary algorithms. The agent proposes black-box optimisation algorithms, and Lean guards filter out flawed designs. The agent then defines a potential function and proves drift bounds that guarantee an expected runtime improvement over the baseline algorithm. At no point in the pipeline are the algorithms compared empirically. The agent generates a black-box optimisation algorithm with expected runtime $O(n^2)$ on a CycleUncover problem, where the classical (1+1) EA needs runtime $e^{Ω(n)}$. The pipeline also designs and analyses a black-box optimisation algorithm for integer-valued submodular minimisation with expected runtime $O(n^{2+2c})$, and it matches the optimal order $O(n^2)$ on LeadingOnes. Across 8 problem classes, 12 of 15 runs with frontier models complete the pipeline with a machine-checked expected-runtime bound. An ablation study shows that removing the staging and asking the agent to zero-shot an algorithmic design with a runtime guarantee leads to a copy of the seeded baseline algorithm (random search) or to incomplete proofs. Our study suggests that machine-checked specifications in Lean can replace empirical evaluation in open-ended algorithm discovery. To the best of our knowledge, this is the first example of an automated pipeline that designs efficient black-box optimisation algorithms with machine-checked runtime bounds in Lean.

Large language models (LLMs) enable automated algorithm design (AAD) through reasoning and code synthesis. However, most existing LLM-based AAD methods separate algorithm design from target optimization, deploying a fixed design even as the optimization state evolves. Conventional adaptive optimizers can respond to such changes, but their adjustments remain confined to predefined parameters, operators, or strategies. To address these limitations, we introduce online LLM-based AAD, a novel optimization paradigm that treats the algorithm itself as a state-dependent decision variable. At each stage, LLM agents synthesize an algorithm with new behavior logic from the current optimization state. Executing the generated algorithm advances the search and provides feedback for subsequent designs, coupling algorithm design with target optimization without requiring a separate offline algorithm pretraining stage. To implement this paradigm, we propose OnDesign, a multi-agent framework that reconciles competing design perspectives to synthesize executable algorithms and uses execution feedback to refine how runtime evidence is interpreted for subsequent designs. We evaluate OnDesign across two mainstream black-box optimization paradigms on three scenarios: Bayesian optimization, evolutionary continuous optimization, and evolutionary mixed-variable optimization. Extensive experiments on six benchmark suites and one engineering problem across multiple problem dimensions demonstrate superior overall performance over conventional optimizers and offline LLM-based AAD methods.