The multi-objective multi-item transportation problem is a challenging issue in the context of supply chain management which deals with optimizing several conflicting objectives, considering the allocation of different products departing from many source nodes to multiple demand destinations. In this paper we propose a systematic mathematical approach based on linear programming to solve this challenging optimization problem. Based on these assumptions the study designs a multi-objective linear programming (MOLP) model with cost, delivery time and environment as the main objectives. The model is developed under clear-cut restrictions that consider supply avai lability, demand requirements, vehicle capacity and multi-product allocation rules. A practical example is considered with real operational data of a regional distribution network for optimal transportation planning and WinQSB software is used to find the best routes. Results show that the proposed model can effectively compromise conflicting multi-objectives, reducing total cost by 18.5%, delivery time by 12.3% and CO2 emissions by 15.2%. The research uses the weighted sum-constraint method for Pareto optimization based decisiontrade-offs, and results into a full tradeoffs analysis and possible transportation planning solutions to decision-making people.

Optimal Transport Barycentric Aggregation for Byzantine-Resilient Federated Learning

Quantum optimal transport with convex regularization

scREPA: Predicting single-cell perturbation responses with cycle-consistent representation alignment.

A fast, large-scale optimal transport algorithm for holographic beam shaping

Chemical doping is one of the promising approaches for tailoring the electronic properties of graphene/copper (Gr/Cu) composites. However, the diversity of doping elements and their complex bonding configurations result in nuanced effects, such as the competition between increased carrier concentration and defect generation (lattice distortion). Therefore, it is essential to decouple such complex doping effects into the intrinsic contribution of the dopant atom and the extrinsic effects of defects like vacancies. In this work, first-principles calculations, deformation potential theory, and the parabolic band model are combined to investigate the intrinsic mechanisms of various dopants. This approach decouples their contributions to carrier concentration and mobility, enabling the effective selection of dopants with optimal carrier transport properties. The corresponding results reveal that dopants which significantly distort the Dirac cone structure, such as O, S, P, Br, and Si, lead to significant degradation of carrier mobility and are thus excluded. In contrast, N is identified as the optimal dopant of Gr/Cu composites, outperforming B by effectively enhancing carrier concentration while well maintaining high carrier mobility, thereby achieving a superior balance for enhanced conductivity. This work establishes a theoretical framework for dopant selection and provides key insights for the design of high-conductivity Gr/Cu composites.

This short paper presents a general approach for computing robust Wasserstein barycenters[2], [80], [81] of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the barycenter and the persistence diagrams. However, this procedure only works for the transportation cost related to the $q$-Wasserstein distance $W_{q}$ when $q=2$. We adapt an alternative fixed-point method[76] to compute a barycenter diagram for generic transportation costs ($q \gt 1$), in particular those robust to outliers , $q \in (1,2)$. We show the utility of our work in two applications : (i) the clustering of persistence diagrams on their metric space and (ii) the dictionary encoding of persistence diagrams [73]. In both scenarios, we demonstrate the added robustness to outliers provided by our generalized framework. Our Python implementation is available at this address: https://github.com/Keanu-Sisouk/RobustBarycenter.

We introduce generalized Frobenius–Perron operators associated with measurable transformations between two measurable spaces and study their basic properties. The new concept extends the well-known Frobenius–Perron operators from ergodic theory to a more general setting. We then discuss the optimal transport problem in terms of generalized Frobenius–Perron operators, providing a new perspective for their theoretical and numerical studies.

Quantifying differences between flow fields is a key challenge in fluid mechanics, particularly when evaluating the effectiveness of flow control or other problem parameters. Traditional vector metrics, such as the Euclidean distance, provide straightforward pointwise comparisons but can fail to distinguish distributional changes in flow fields. To address this limitation, we employ optimal transport (OT) theory, which is a mathematical framework built on probability and measure theory. By aligning Euclidean distances between flow fields in a latent space learned by an autoencoder with the corresponding OT geodesics, we seek to learn low-dimensional representations of flow fields that are interpretable from the perspective of unbalanced OT. As a demonstration, we utilise this OT-based analysis on separated flows past a NACA 0012 airfoil with periodic heat flux actuation near the leading edge. The cases considered are at a chord-based Reynolds number of 23 000 and a free-stream Mach number of 0.3 for two angles of attack (AoA) of $6^\circ$ and $9^\circ$ . For each angle of attack, we identify a two-dimensional embedding that succinctly captures the different effective regimes of flow responses and control performance, characterised by the degree of suppression of the separation bubble and secondary effects from laminarisation and trailing-edge separation. The interpretation of the latent representation was found to be consistent across the two AoA, suggesting that the OT-based latent encoding was capable of extracting physical relationships that are common across the different suites of cases. This study demonstrates the potential utility of optimal transport in the analysis and interpretation of complex flow fields.

Graph invariant learning (GIL) seeks invariant relations between graphs and labels under distribution shifts. Recent works try to extract an invariant subgraph to improve out-of-distribution (OOD) generalization, yet existing approaches either lack explicit control over compactness or rely on hard top-$k$ selection that shrinks the solution space and is only partially differentiable. In this paper, we provide an in-depth analysis of the drawbacks of some existing works and propose a few general principles for invariant subgraph extraction: 1) separability, as encouraged by our sparsity-driven mechanism, to filter out the irrelevant common features; 2) softness, for a broader solution space; and 3) differentiability, for a soundly end-to-end optimization pipeline. Specifically, building on optimal transport, we propose Graph Sinkhorn Attention (GSINA), a fully differentiable, cardinality-constrained attention mechanism that assigns sparse-yet-soft edge weights via Sinkhorn iterations and induces node attention. GSINA provides explicit controls for separability and softness, and uses a Gumbel reparameterization to stabilize training. It convergence behavior is also theoretically studied. Extensive empirical experimental results on both synthetic and real-world datasets validate its superiority.

Deep learning-based feature matching has showcased great superiority for point cloud registration. While coarse-to-fine matching architectures are prevalent, they typically perform sparse and geometrically inconsistent coarse matching. This forces the subsequent fine matching to rely on computationally expensive optimal transport and hypothesis-and-selection procedures to resolve inconsistencies, leading to inefficiency and poor scalability for large-scale real-time applications. In this paper, we design a consistency-aware spot-guided Transformer (CAST) to enhance the coarse matching by explicitly utilizing geometric consistency via two key sparse attention mechanisms. First, our consistency-aware self-attention selectively computes intra-point-cloud attention to a sparse subset of points with globally consistent correspondences, enabling other points to derive discriminative features through their relationships with these anchors while propagating global consistency for robust correspondence reasoning. Second, our spot-guided cross-attention restricts cross-point-cloud attention to dynamically defined "spots"-the union of correspondence neighborhoods of a query's neighbors in the other point cloud, which are most likely to cover the true correspondence of the query ensured by local consistency, eliminating interference from similar but irrelevant regions. Furthermore, we design a lightweight local attention-based fine matching module to precisely predict dense correspondences and estimate the transformation. Extensive experiments on both outdoor LiDAR datasets and indoor RGB-D camera datasets demonstrate that our method achieves state-of-the-art accuracy, efficiency, and robustness. Besides, our method showcases superior generalization ability on our newly constructed challenging relocalization and loop closing benchmarks in unseen domains. Our code and models are available at https://github.com/RenlangHuang/CASTv2.

Transformer architecture has shown significant potential in various visual tasks, including point cloud registration. Positional encoding, as an order-aware module, plays a crucial role in Transformer framework. In this paper, we propose OIF-PCR++, a conditional positional encoding (CPE) method for point cloud registration. The core CPE module utilizes length and vector encoding at different stages, conditioned on the relative pose states between the point clouds to be registered. As a result, it progressively alleviates the feature ambiguity through the incorporation of geometric cues. Building upon the proposed CPE, we introduce an iterative positional encoding optimization pipeline comprising two stages: 1) We find one correspondence via a differentiable optimal transport layer, and use it to encode length information into the point cloud features, which alleviates challenges arising from differing reference frames by enhancing spatial consistency. 2) We apply a progressive direction alignment strategy to achieve rough alignment between the paired point clouds, and then gradually incorporate direction information with the aid of this alignment, further enhancing feature distinctiveness and reducing feature ambiguity. Through this iterative optimization process, length and direction information are effectively integrated to achieve consistent and distinctive positional encoding, thus enabling the learning of discriminative point cloud features. Additionally, we present an inlier propagation mechanism that harmoniously integrates consistent geometric information for positional encoding. The proposed positional encoding is highly efficient, introducing only a marginal increase in computational overhead while significantly improving feature distinguishability. Extensive experiments demonstrate that our method achieves superior performance compared to state-of-the-art methods across indoor, outdoor, object-level, and multi-way benchmarks, while also generalizing well to complex real-world scenarios. The code will be released upon acceptance to support the research community.

Kantorovich type optimal transportation problems with nonlinear cost functions are treated, including the dependence on conditional measures of transport plans. A range of nonlinear Kantorovich problems for cost functions of a special form is considered and results on the existence (or nonexistence) of optimal solutions are proved. The relationshp is established between the nonlinear Kantorovich problem with the cost function of some special form and the Monge problem with convex dominance.

Federated Domain Generalization (FedDG) aims to train a globally generalizable model on data from decentralized, heterogeneous clients. While recent work has adapted vision-language models for FedDG using prompt learning, the prevailing "one-prompt-fits-all" paradigm struggles with sample diversity, causing a marked performance decline on personalized samples. The Mixture of Experts (MoE) architecture offers a promising solution for specialization. However, existing MoE-based prompt learning methods suffer from two key limitations: coarse image-level expert assignment and high communication costs from parameterized routers. To address these limitations, we propose TRIP, a Token-level pRompt mIxture with Parameter-free routing framework for FedDG. TRIP treats prompts as multiple experts, and assigns individual tokens within an image to distinct experts, facilitating the capture of fine-grained visual patterns. To ensure communication efficiency, TRIP introduces a parameter-free routing mechanism based on capacity-aware clustering and Optimal Transport (OT). First, tokens are grouped into capacity-aware clusters to ensure balanced workloads. These clusters are then assigned to experts via OT, stabilized by mapping cluster centroids to static, non-learnable keys. The final instance-specific prompt is synthesized by aggregating experts, weighted by the number of tokens assigned to each. Extensive experiments across four benchmarks demonstrate that TRIP achieves optimal generalization results, with communicating as few as 1K parameters. Our code is available at https://github.com/GongShuai8210/TRIP.

Strong c-concavity and stability in optimal transport

Visual reinforcement learning via sequential consistency preserved policy contrast from optimal transport view.

The whole-body pose estimation task aims to predict the location of keypoints of the face, body, hands, and feet given an image. However, scale variation in different parts of the human body and semantic ambiguity in small-scale parts cause performance degradation in keypoint localization. The traditional paradigm for solving multiscale issues is to construct multiscale feature representations. Nevertheless, multiscale features extracted from visual images do not eliminate the semantic ambiguity issue in the small-scale part. In this article, we propose affiliation alignment network (A2Net), which solves the aforementioned problem by alignment of vision-language hierarchical affiliations. Specifically, text modality has the advantage of not being affected by the scaling problem and the small-scale semantic ambiguity problem, which is due to image scale variations. We construct a multisemantic hierarchical language latent space with clear semantic and affiliation relations by designing Text Affiliation Injection operations. Subsequently, we adopt the optimal transport (OT) method to align image features of different scales with text features of the corresponding hierarchical levels to build an image scale-independent visual-language latent space, which overcomes the image scale problem and the small-scale semantic ambiguity problem. Extensive experimental results on two whole-body pose estimation datasets show that our model achieves convincing performance compared to the current state-of-the-art methods. The code is openly available at https://github.com/LingLin-ll/A2Net.

This study investigates the AZO/Cu/Ag/AZO multilayer thin films deposited on PET substrates via RF confocal magnetron sputtering, focusing on the influence of Cu and Ag interlayer thickness ratios, while maintaining a constant total metal thickness of 10 nm, on their microstructural, optical, and electrical properties. X-ray diffraction (XRD) analysis combined with Rietveld refinement revealed that Ag-rich configurations enhance crystallinity, reduce residual stress, and promote a pronounced (002) preferential orientation in the AZO layers, which is attributed to improved nucleation behavior and reduced interfacial disorder. Atomic force microscopy (AFM) analysis demonstrated that the Cu/Ag thickness ratio effectively modulates nanoparticle coalescence and surface roughness. Optical characterization revealed that Ag-rich films exhibit higher visible transmittance owing to improved film continuity and constructive interference effects, whereas Cu-rich structures favor enhanced near-infrared (NIR) transparency. Electrical properties, evaluated using Hall effect measurements, indicated that a balanced Cu/Ag configuration (5/5 nm) delivers optimal charge transport, achieving the lowest resistivity (1.22 × 10 −3 Ω.cm), the highest carrier mobility (7.08 cm 2 .V −1 .s −1 ), and the maximum figure of merit (6.88 × 10 −5 Ω −1 ). These results demonstrate that rational Cu/Ag interlayer engineering enables an effective balance between transparency and conductivity in flexible transparent electrodes, opening viable pathways for applications in wearable electronics, flexible photovoltaics, and foldable display technologies.

Optimal mobile energy transportation: A co-optimization approach

Unsupervised deep hashing based on multi-scale aggregation and optimal transport matching for image retrieval