Let A be the map from RX to RX Y that sends a function f on X to the function f(x) p P(x,y)onX Y. Similarly,let B bethemapfrom RY toRX Y thatsendsafunction gto g(y) p P(x,y). Combining these, we get exactly the stated formula. Here we elaborate further on COPT's optimization routine. As the objective Equation 3.1 is not globally convex, gradient descent can fall into local minima. But this requires anontrivialnumber (e.g.

Weintroduce COPT,anoveldistance metric between graphs defined via anoptimization routine, computing a coordinated pair of optimal transport maps simultaneously. This gives an unsupervised way to learn general-purpose graph representation, applicable tobothgraphsketching andgraphcomparison.

The integration of Large Language Models (LLMs) with optimization modeling offers a promising avenue for advancing decision-making in operations research (OR). Traditional optimization methods,such as linear programming, mixed integer programming, and simulation depend heavily on domain expertise to translate real-world problems into solvable mathematical models. While solvers like Gurobi and COPT are powerful, expert input remains essential for defining objectives, constraints, and variables. This research investigates the potential of LLMs, specifically the DeepSeek-R1 model, to bridge this formulation gap using natural language understanding and code generation. Although prior models like GPT-4, Claude, and Bard have shown strong performance in NLP and reasoning tasks, their high token costs and tendency toward hallucinations limit real-world applicability in supply chain contexts. In contrast, DeepSeek-R1, a cost-efficient and high-performing model trained with reinforcement learning, presents a viable alternative. Despite its success in benchmarks such as LiveCodeBench and Math-500, its effectiveness in applied OR scenarios remains under explored. This study systematically evaluates DeepSeek-R1 across four key OR benchmarks: NL4OPT, IndustryOR, EasyLP, and ComplexOR. Our methodology includes baseline assessments, the development of a hallucination taxonomy, and the application of mitigation strategies like LLM-as-a-Judge, Few-shot Learning (FSL), Tool Calling, and a Multi-agent Framework. These techniques aim to reduce hallucinations, enhance formulation accuracy, and better align model outputs with user intent.

This paper introduces a two-timescale hierarchical decentralized control architecture for Cyber-Physical Systems (CPS). The system consists of a global controller (GC), and N local controllers (LCs). The GC operates at a slower timescale, imposing budget constraints on the actions of LCs, which function at a faster timescale. Applications can be found in energy grid planning, wildfire management, and other decentralized resource allocation problems. We propose and analyze two optimization frameworks for this setting: COpt and FOpt. In COpt, both GC and LCs together optimize infinite-horizon discounted rewards, while in FOpt the LCs optimize finite-horizon episodic rewards, and the GC optimizes infinite-horizon rewards. Although both frameworks share identical reward functions, their differing horizons can lead to different optimal policies. In particular, FOpt grants greater autonomy to LCs by allowing their policies to be determined only by local objectives, unlike COpt. To our knowledge, these frameworks have not been studied in the literature. We establish the formulations, prove the existence of optimal policies, and prove the convergence of their value iteration algorithms. We further show that COpt always achieves a higher value function than FOpt and derive explicit bounds on their difference. Finally, we establish a set of sufficient structural conditions under which the two frameworks become equivalent.

In this supplement, we give full proofs of Lemmas 3.1 and 3.2, further discuss COPT optimizations We give full proofs to Lemma 3.1, an analytic formula for the COPT metric, and Lemma 3.2, the Combining these, we get exactly the stated formula. Symmetry is easiest to check using Lemma 3.1 and the fact that the trace of the square root of a Finally, we check the triangle inequality. Here we elaborate further on COPT's optimization routine. As the objective Equation 3.1 is not globally convex, gradient descent can fall into local minima. This is repeated 20 times.

We introduce COPT, a novel distance metric between graphs defined via an optimization routine, computing a coordinated pair of optimal transport maps simultaneously. This is an unsupervised way to learn general-purpose graph representations, it can be used for both graph sketching and graph comparison. COPT involves simultaneously optimizing dual transport plans, one between the vertices of two graphs, and another between graph signal probability distributions. We show both theoretically and empirically that our method preserves important global structural information on graphs, in particular spectral information, making it well-suited for tasks on graphs including retrieval, classification, summarization, and visualization.