Generating feasible Pareto fronts for constrained bi-objective continuous optimization is central to multi-criteria decision-making. Existing methods usually rely on iterative scalarization, evolutionary search, or problem-specific solvers, requiring repeated optimization for each instance. We introduce DIPS, an end-to-end framework that fine-tunes large language models as amortized Pareto-front generators for constrained bi-objective convex optimization. Given a textual problem description, DIPS directly outputs an ordered set of feasible continuous decision vectors approximating the Pareto front. To make continuous optimization compatible with autoregressive language modeling, DIPS combines a compact discretization scheme, Numerically Grounded Token Initialization for new numerical tokens, and Three-Phase Curriculum Optimization, which progressively aligns structural validity, feasibility, and Pareto-front quality. Across five families of constrained bi-objective convex problems, a fine-tuned 7B-parameter model achieves normalized hypervolume ratios of 95.29% to 98.18% relative to reference fronts. With vLLM-accelerated inference, DIPS solves one instance in as little as 0.16 seconds and outperforms general-purpose and reasoning LLM baselines under the evaluated setting. These results suggest that LLMs can serve as effective amortized generators for continuous Pareto-front approximation.
Achieving strong optimization generalization across diverse optimization problems while requiring limited training resources remains a challenging problem for optimization-oriented large language models (LLMs). Existing approaches typically rely on large-scale supervised datasets, costly reasoning annotations, and expensive intermediate step verification, resulting in substantial training overhead. To address these challenges, we propose MiniOpt, a reinforcement learning framework that learns to solve optimization problems through an "reasoning-to-model-and-solve" paradigm. MiniOpt decomposes optimization reasoning into structured optimization modeling and executable solver generation. Building upon this paradigm, we introduce OptReward, a reward function with hierarchical score structure that jointly evaluates formulation and solution, enabling effective policy learning without expert demonstrations. We further develop an optimization-oriented policy optimization strategy that improves exploration efficiency and stabilizes reinforcement learning for compact models. Extensive experiments show that MiniOpt-3B exhibits strong optimization generalization across various optimization types, problem scenarios, and task domains. For models with fewer than 10B parameters, MiniOpt series achieves the highest average solving accuracy (SA). For models with more than 10B parameters, MiniOpt still shows competitive performance. These results suggest that optimization-oriented reward design and reinforcement learning provide an effective pathway for developing compact optimization-specialized language models with strong optimization generalization capabilities. The code is available at https://github.com/Hsiang-1/MiniOpt.
Mathematical optimization is a powerful tool for structured decision-making across domains such as resource allocation and planning. Formulating optimization models faithful to reality, though, remains a significant bottleneck as it typically demands both domain expertise and optimization knowledge that are often scarce. Recent advances in large language models (LLMs) promise to bridge this gap, enabling the generation of candidate optimization models from natural language descriptions. However, there is no guarantee that any single LLM-generated model is reliable, and existing approaches that output only one model are therefore risky. In this work, we propose a novel algorithm that generates a portfolio of optimization models, designed to be robust to the limitations of LLMs. Our method exploits the observation that a single LLM can play two distinct roles \unicodex2014 as a stochastic generator and as a reasoning evaluator \unicodex2014 and proposes a unified framework that leverages both capabilities in a complementary manner. We provide theoretical guarantees showing that, as long as either the generator or the evaluator is well-aligned with human preferences, the portfolio is guaranteed to contain high-quality candidates, enabling a principled human-in-the-loop process in which a decision-maker can review multiple candidates before committing to one. We further validate our approach empirically, demonstrating strong performance across a range of optimization modeling tasks.
Large language models are increasingly required to generate responses that satisfy multiple competing objectives. Since optimal trade-offs depend on both user preferences and input prompts, controllable multi-objective generation must dynamically adapt models at inference time without retraining. To address this, we propose Evolutionary Soups, a mixture-of-experts framework for fine-grained generation control, with gating networks trained via an evolutionary algorithm. The per-layer gating networks dynamically produce expert-merging coefficients from hidden-state representations, while the evolutionary algorithm incorporates greedy hypervolume contribution for effective evolution of these gating networks, achieving consistent improvements on large and noisy training datasets and broader coverage of the non-convex Pareto front. Experiments across three tasks demonstrate the effectiveness of Evolutionary Soups over baselines: it achieves the best hypervolume, linear utility, and Tchebyshev utility (~20% improvement) among controllable methods on all tasks.