Beyond Objective Equivalence: Constraint Injection for LLM-Based Optimization Modeling on Vehicle Routing Problems

Large language models (LLMs) increasingly translate natural-language optimization problems into executable solver code. Yet for constraint-dense operations research (OR) problems, existing data-filtering and training pipelines largely rely on objective-equivalence signals such as differential testing and answer agreement, which a program can pass while adding spurious constraints or silently omitting required ones, whenever those constraints are non-binding on the tested instance. We propose constraint injection, which uses feasible probes to expose spurious over-constraint and one-constraint-violating probes to reveal silent constraint omission. Combined with differential testing, it forms a dual verifier. We instantiate and evaluate it on vehicle routing problems (VRPs), a representative constraint-dense combinatorial optimization testbed with coupled operational constraints. We develop VRPCoder, an 8B end-to-end model that translates natural-language VRP scenarios into Gurobi scripts, together with an expert-verified VRP benchmark suite covering 21 variants. The verifier is reused as a rejection-sampling filter during data synthesis and as a per-rollout reward in group relative policy optimization (GRPO). Across four VRP benchmarks, VRPCoder-GRPO reaches 93\% average Pass@1, outperforms Gemini-3.1-Pro Preview on three benchmarks, exceeds Claude-Sonnet-4.5 by 28 average points, and surpasses prior OR-LLMs by 78 average points.

Interpretable Logical Anomaly Classification via Constraint Decomposition and Instruction Fine-Tuning

Logical anomalies are violations of predefined constraints on object quantity, spatial layout, and compositional relationships in industrial images. While prior work largely treats anomaly detection as a binary decision, such formulations cannot indicate which logical rule is broken and therefore offer limited value for quality assurance. We introduce Logical Anomaly Classification (LAC), a task that unifies anomaly detection and fine-grained violation classification in a single inference step. To tackle LAC, we propose LogiCls, a vision-language framework that decomposes complex logical constraints into a sequence of verifiable subqueries. We further present a data-centric instruction synthesis pipeline that generates chain-of-thought (CoT) supervision for these subqueries, coupling precise grounding annotations with diverse image-text augmentations to adapt vision language models (VLMs) to logic-sensitive reasoning. Training is stabilized by a difficulty-aware resampling strategy that emphasizes challenging subqueries and long tail constraint types. Extensive experiments demonstrate that LogiCls delivers robust, interpretable, and accurate industrial logical anomaly classification, providing both the predicted violation categories and their evidence trails.