Multi-Objective Hierarchical Optimization with Large Language Models

TL;DR

MOHOLLM addresses black-box multi-objective optimization by integrating Large Language Models as both region-specific samplers and surrogates within a hierarchical, axis-aligned space-partitioning framework. It adaptively splits the input domain into disjoint hyperrectangles, scores regions with a composite metric that blends regional Hypervolume contribution, geometric size, and uncertainty, and uses the LLM to generate localized candidates and predict objective values, selecting batches for true evaluation to maximize HV. The authors prove almost-sure convergence of the sampled points to the true Pareto set in Hausdorff distance $d_H$, and HV consistency under standard assumptions. Empirically, MOHOLLM outperforms a global LLM baseline and is competitive with state-of-the-art MOBO and MOEAs across 15 synthetic and real-world benchmarks, with extensive ablations demonstrating the importance of LLM size, prompt design, and surrogate capabilities.

Abstract

Despite their widespread adoption in various domains, especially due to their powerful reasoning capabilities, Large Language Models (LLMs) are not the off-the-shelf choice to drive multi-objective optimization yet. Conventional strategies rank high in benchmarks due to their intrinsic capabilities to handle numerical inputs and careful modelling choices that balance exploration and Pareto-front exploitation, as well as handle multiple (conflicting) objectives. In this paper, we close this gap by leveraging LLMs as surrogate models and candidate samplers inside a structured hierarchical search strategy. By adaptively partitioning the input space into disjoint hyperrectangular regions and ranking them with a composite score function, we restrict the generative process of the LLM to specific, high-potential sub-spaces, hence making the problem easier to solve as the LLM doesn't have to reason about the global structure of the problem, but only locally instead. We show that under standard regularity assumptions, our algorithm generates candidate solutions that converge to the true Pareto set in Hausdorff distance. Empirically, it consistently outperforms the global LLM-based multi-objective optimizer and is on par with standard evolutionary and Bayesian optimization algorithm on synthetic and real-world benchmarks.

Figures (21)

• Figure 1: Illustration of MOHOLLM's optimization pipeline.
• Figure 2: Search process of MOHOLLM on the 2D Branin–Currin function at different optimization stages. The last column illustrates the function values, while other columns the five algorithmic steps: partitioning, region scoring, probabilistic selection, LLM-based sampling, and evaluation. Rows correspond to early, intermediate, and late trials. MOHOLLM transitions from coarse, globally exploratory partitions to fine-grained, localized refinement around the Pareto front, allocating samples to both sparse regions and dense Pareto-optimal areas.
• Figure 3: Hypervolume (HV) over function evaluations on synthetic benchmarks (DTLZ1, Branin-Currin, Chankong–Haimes, SchafferN1 and Kursawe). MOHOLLM and the LLM baseline curves are marked to highlight them further.
• Figure 4: Hypervolume (HV) of MOHOLLM and baselines over function evaluations on real-world benchmarks (Penicillin, Vehicle Safety, and Car Side Impact).
• Figure 5: HV trajectories of MOHOLLM with different LLM surrogates and samplers.

Theorems & Definitions (9)

• Definition 2.1
• Definition 2.2
• Definition 2.3: Hypervolume Indicator
• Lemma 4.6: The Shrinking Lemma
• proof
• Theorem 4.7: Almost-Sure Pareto Consistency
• proof
• Corollary 4.8: Hypervolume Consistency
• proof