Convex Hull Research Articles

Energy management in smart distribution networks: Synergizing network reconfiguration, energy storage, and electric vehicles with disjunctive convex hull relaxation

Efficient energy management is critical for modern distribution networks integrating renewable energy, storage systems, and electric vehicles. This paper introduces a novel approach combining network reconfiguration with disjunctive convex hull relaxation to optimize power flow, minimize energy losses, and enhance grid stability. Case studies on 33-node and 84-node networks showcase the model’s effectiveness, reducing energy losses to 0.939 MWh and 7.214 MWh, respectively, outperforming Big-M and McCormick methods. Additionally, the approach achieved a 78 % efficiency improvement, enhanced emissions control, and improved voltage regulation, demonstrating superior renewable energy integration. These results highlight the model's potential to provide a more efficient and sustainable energy management solution, addressing the challenges of modern power grids.

A Polyhedral Study of Multivariate Decision Trees

Decision trees are a widely used tool for interpretable machine learning. Multivariate decision trees employ hyperplanes at the branch nodes to route datapoints throughout the tree and yield more compact models than univariate trees. Recently, mixed-integer programming (MIP) has been applied to formulate the optimal decision tree problem. A key component of most MIP formulations is a specification of how to route datapoints in the tree from the root to the leaves. Our goal is to provide polyhedral characterizations of realizable routings, that is, routings that can be realized using multivariate branching rules. We first focus on shattering inequalities, a class of valid inequalities that can be used to strengthen almost any MIP formulation and that have been shown to be computationally effective. We prove that if all the feature vectors are in general position, then the shattering inequalities defined for the root of the tree are facet defining for the convex hull of the realizable routings. We then show that every facet-defining inequality of a depth one tree involving at least two variables is also facet defining for trees of arbitrary depth. Finally, we show that facet-defining inequalities characterizing realizable routings are also facet-defining for a complete MIP formulation. We validate our theoretical results by performing numerical experiments that specifically exploit shattering inequalities defined at the root node and we observe an improvement in performance for both numerical and categorical data sets, especially for decision trees of intermediate size. Funding: C. Michini gratefully acknowledges funding from the DoD, Air Force [Award FA9550-23-1-0487].

Computing the minimal perimeter polygon for digital objects in the triangular tiling

This work presents an algorithm, together with its correctness proof, to determine the minimum perimeter polygon (MPP) for digital objects given as regular complexes in the triangular plane tiling. Such objects are edge-adjacency-connected sets of triangle tiles that have no end tiles, and the point set union of all their tiles forms a simple polygon. Nevertheless, the boundary paths of the objects are not assumed to be simple. Then the MPP is a weakly simple polygon that coincides with the relative convex hull (i.e., geodesic hull) of a set A with respect to a simple polygon B, where A⊂B, but A is not necessarily a polygon, in fact it is generally not connected. Our MPP algorithm relies on constructing and iteratively constraining cones of visibility through forthcoming boundary tiles, it uses the structure of the canonical boundary path, the MPP frontier is the shortest polygonal curve following this path. We also propose a boundary tracing algorithm to obtain such paths from the objects.

HBSP: a hybrid bilinear and semidefinite programming approach for aligning partially overlapping point clouds

In many applications, there is a need for algorithms that can align partially overlapping point clouds while remaining invariant to corresponding transformations. This research presents a method that achieves these goals by minimizing a binary linear assignment-least squares (BLALS) energy function. First, we reformulate the BLALS problem as the minimization of a quadratic function with quadratic and linear constraints through variable substitution. By utilizing semidefinite relaxation and the convex envelope of bilinear monomials, we relax the problem to create a lower bound that can be solved using linear assignment and low-dimensional semidefinite programming. Additionally, we develop a branch-and-bound (BnB) algorithm that only branches over the transformation variable, which enhances convergence. Experimental results show that, compared to state-of-the-art approaches, the proposed method is robust against non-rigid deformation and outliers when the outliers are separate from the inliers. However, its robustness decreases when outliers are mixed with inliers. The run time of our method is relatively high due to the need to solve a semidefinite program in each iteration of the BnB algorithm.

Video tracking of single cells to identify clustering behavior

Cancer cell clustering is a critical factor in metastasis, with cells often believed to migrate in groups as they establish themselves in new environments. This study presents preliminary findings from an in vitro experiment, suggesting that co-culturing cells provides an effective method for observing this phenomenon, even though the cells are grown as monolayers. We introduce a novel single-cell tracking approach based on graph theory to identify clusters in PC3 cells cultivated in both monoculture and co-culture with PC12 cells, using 66-h time-lapse recordings. The initial step consists of defining “linked” pairs of PC3 cells, laying the foundation for the application of graph theory. We propose two alternative definitions for cell pairings. The first method, Method 1, defines cells as “linked” at a given time t if they are close together within a defined time period before and after t. A second potential alternative method, Method 2, pairs cells if there is an overlap between the convex hulls of their respective tracks during this time period. Pairing cells enables the application of graph theory for subsequent analysis. This framework represents a cell as a vertex (node) and a relation between two cells as an edge. An interconnected set of high-degree nodes (nodes with many connections or edges) forms a subgraph, or backbone, that defines a patch (cluster) of cells. All nodes connected to this backbone are part of the subgraph. The backbone of high-degree nodes functions as a partition (or cut) of the initial graph. Two consecutive clusters in the video are considered to share the same identity if the following cluster contains at least p = 75 % of the cells from the preceding cluster, and the mean positions of their cells are within △r = 75μm. PC3 cells grown in co-culture appear to form persistent clusters exceeding 10 cells after 40–50 h incubation following seeding. In contrast, PC3 cells cultured alone (mono-culture) did not exhibit this behavior. This approach is experimental and requires further validation with a broader dataset.

Distributed Adaptive Predefined-Time Bipartite Containment Algorithm for Nonlinear Multi-Agent Systems with Actuator Faults

Distributed adaptive predefined-time bipartite containment for a class of second-order nonlinear multi-agent systems are studied with actuator faults. The communication topology of multi-agent systems is fixed and directed. To ensure that followers can reach the convex hull spanned by leaders under the conditions of actuator faults, the sliding mode method is introduced. Control protocol for multi-agent systems demonstrates its effectiveness. Finally, simulations are provided to verify the effectiveness of the proposed algorithm.

Introducing a simple convex hull method to calibrate diffusion coefficients in Lagrangian particle models

Calibrating Lagrangian particle models (LPMs) is crucial for simulating oil spill trajectories, enabling emergency responses, and mitigating harmful effects on the environment. However, there is a lack of rigorous observation data on oil spill events, especially in water bodies with no serious spills but perceived increasing risk, hampering LPM application. This study utilized trajectory data from three drifters in Lake Erie, collected in 2016, 2018, and 2019. We then proposed a simple and efficient convex hull method to identify the smallest envelope of the simulated particle cloud, covering all drifter trajectories while simultaneously controlling over-diffusion. This approach was employed to calibrate the General NOAA Operational Modeling Environment oil spill model (an LPM) and determine optimal diffusion coefficients. Based on 362 one-day-long drifter trajectory simulation cases, the recommended diffusion coefficient for particle trajectory simulation in Lake Erie was determined to be 14 m2 s−1. The calibrated model demonstrated the ability to effectively capture the movement direction of drifters and control the discrepancy between simulated and observed trajectories. Meanwhile, it underestimated drifter travel distances due to errors in input currents, the primary force driving drifter movement. These results promote LPM application for particle trajectory simulations.

Algorithmic Efficiency in Convex Hull Computation: Insights from 2D and 3D Implementations

This study examines various algorithms for computing the convex hull of a set of n points in a d-dimensional space. Convex hulls are fundamental in computational geometry and are applied in computer graphics, pattern recognition, and computational biology. Such convex hulls can also be useful in symmetry problems. For instance, when points are arranged symmetrically, the convex hull is also likely to be symmetrically shaped, which can be useful for object recognition in computer vision or pattern recognition. The focus is primarily on two-dimensional algorithms, including well-known methods like Gift Wrapping, Graham Scan, Divide and Conquer, QuickHull, TORCH, Kirkpatrick–Sediel, and Chan’s algorithms. These algorithms vary in terms of time complexity and scalability to higher dimensions. This study is extended to three-dimensional convex hull algorithms, such as NAW, randomized insertion, and parallelized versions, such as CudaHull and CudaChain. This study aimed to elucidate the operational principles, step-by-step procedures, and comparative time complexities of each algorithm. The implementation in Python facilitates a detailed comparison of the algorithmic performance through stepwise analysis and graphical outputs. The ultimate goal is to provide insights into the strengths and weaknesses of each algorithm under various scenarios, thereby offering a comprehensive guide for practical implementation.

Seasonal variation in home-range size of the White-backed Woodpecker

AbstractKnowing a species’ area requirements is fundamental for species conservation. For the nominate subspecies of the White-backed Woodpecker Dendrocopos leucotos, a species of high conservation concern in Europe, estimates of the seasonal and year-round area requirements based on telemetry are missing. In the present study, we radio-tracked adult White-backed Woodpeckers in Central Europe and investigated bi-monthly home-range sizes based on three home-range estimators in relation to season, sex, body weight, and year. Home-range size of 49 radio-tracked individuals varied depending on the used home-range estimator, with minimum convex polygons (MCP) and autocorrelated kernel density estimation (AKDE) producing 1.6–1.8 and 2–3.3 times larger seasonal home ranges than traditional kernel density estimation (KDE). Moreover, home-range sizes varied between seasons. Home ranges were smallest in February/March (predicted median home-range sizes ranged from 35 ha with KDE to 88 ha with AKDE) and April/May (KDE: 30 ha, AKDE: 55 ha) and larger during the rest of the year (KDE: 48–67 ha, AKDE: 136–184 ha). The mean home-range size of six individuals tracked in all seasons (calculated with all locations per individual) was 116 ha with KDE, 304 ha with MCP and 350 ha with AKDE. Our results highlight the importance of considering the full annual cycle when addressing area requirements of White-backed Woodpeckers and likely also of other species. Furthermore, our study shows that using multiple methods for home-range estimation may be useful to obtain results that are both comparable with those of other studies and capture the range in which the true home-range size is likely to be. For the conservation of the White-backed Woodpecker, we conclude that at least 116–350 ha of forest should be present for a pair.

DynaHull: Density-centric Dynamic Point Filtering in Point Clouds

In the field of indoor robotics, accurately localizing and mapping in dynamic environments using point clouds can be a challenging task due to the presence of dynamic points. These dynamic points are often represented by people in indoor environments, but in industrial settings with moving machinery, there can be various types of dynamic points. This study introduces DynaHull, a novel technique designed to enhance indoor mapping accuracy by effectively removing dynamic points from point clouds. DynaHull works by leveraging the observation that, over multiple scans, stationary points have a higher density compared to dynamic ones. Furthermore, DynaHull addresses mapping challenges related to unevenly distributed points by clustering the map into smaller sections. In each section, the density factor of each point is determined by dividing the number of neighbors by the volume these neighboring points occupy using a convex hull method. The algorithm removes the dynamic points using an adaptive threshold based on the point count of each cluster, thus reducing the false positives. The performance of DynaHull was compared to state-of-the-art techniques, such as ERASOR, Removert, OctoMap + SOR , and Dynablox, by comparing each method to the ground truth map created during a low activity period in which only a few dynamic points were present. The results indicated that DynaHull outperformed these techniques in various metrics, noticeably in the Earth Mover’s Distance, false negatives and false positives. The data and code for DynaHull are available at https://github.com/Pejman712/DynaHull.git.

Exhaustive search for novel multicomponent alloys with brute force and machine learning

We present an algorithm for the high-throughput computational discovery of intermetallic compounds in systems with a large number of components. It is particularly important for high entropy alloys (HEAs), where multiple principal elements can form numerous potential intermetallic compounds during the condensation process, making it challenging to predict the dominant phase. Our algorithm is based on a brute-force evaluation of candidate structures with a fixed underlying lattice (FCC or BCC) accelerated by machine-learning interatomic potentials. The algorithm takes a set of chemical elements and a crystal lattice type as inputs and produces structures on and near the convex hull of thermodynamically stable structures. The candidate structures are evaluated using the low-rank potential (LRP), trained to reproduce energies of structures equilibrated with density functional theory (DFT). Thanks to extreme computational effectiveness of the LRP, it is feasible to evaluate hundreds of thousands of structures per second, per CPU core. Thus, our algorithm screens a complete set of candidate structures for a given system without missing any configurations. We validated our method on systems with BCC (Nb-W, Nb-Mo-W, V-Nb-Mo-Ta-W) and FCC (Cu-Pt, Cu-Pd-Pt, Cu-Pd-Ag-Pt-Au) lattices and discovered 268 new alloys not reported in the AFLOW database1, which we used as a benchmark.

A high-throughput method for monitoring growth of lettuce seedlings in greenhouses based on enhanced Mask2Former

Monitoring plant growth is crucial for cultivation management. Agronomists can assess the health status of lettuce seedlings based on monitoring results to implement relevant management measures for improving the quality and yield of lettuce seedlings. This study developed a non-destructive, high-throughput growth monitoring method suitable for large-scale assessment of lettuce seedling quality in nurseries. The method utilizes a plant high-throughput phenotyping platform to acquire 10-day time-series imagery data. An Mask2Former network model enhanced by multidimensional collaborative attention mechanism, combined with sliding window and morphological operations, achieves precise recognition and localization of seedling trays, varieties, and individual seedling plants in a progressive manner. Based on individual seedling localization and segmentation results, the method estimates emergence numbers and rates for each variety, and further achieves instance segmentation and counting of individual seedling leaves, innovatively constructing leaf segmentation results of different varieties across the entire seedling tray. Applied to time-series images, the method automatically monitored seedling emergence changes and growth trends for 1,086 lettuce varieties. In monitoring these varieties, the method achieved a coefficient of determination (R2) of 0.96 for emergence number estimation. The extraction of all six key phenotypic parameters demonstrated exceptionally high correlations: projected area, projected perimeter, convex hull area, and convex hull perimeter all showed R2 above 0.99, while leaf compactness R2 was 0.9698, and leaf count R2 was 0.91. Results demonstrate that this high-throughput, reliable method can effectively monitor the growth status of large-scale lettuce seedlings and provide technical support for lettuce nursery quality assessment.

Spatial Ecology of a Resident Avian Predator During the Non-Breeding Period in Managed Habitats of Southeastern Europe.

Describing home range and resource selection is crucial for understanding ecological needs and creating conservation programs. Still, our knowledge of spatial and behavioural ecology for most species remains limited. Here, we used satellite transmitters to investigate core and home range sizes, habitat selection, and roost characteristics in seven tawny owl males in Western Serbia during the autumn-winter period 2023. Using minimum convex polygon (MCP) and autocorrelated kernel density estimation (AKDE), we found clear variability in core area and home range sizes. Also, adult and heavier males have smaller core area and home ranges than juvenile and lightweight individuals. The Bhattacharyya coefficient showed minor home range overlap in tagged males. The final model for evaluation of habitat selection suggests that the likelihood of owl occurrence was positively correlated with the share of anthropogenic infrastructure and negatively associated with the increase in the proportion of cultivated land within the home range. However, scores of model performance metrics showed moderate predictive accuracy, implying that other unmeasured variables may dictate species presence. Our study illustrates the ecological plasticity and ability of the tawny owl to adapt to a human-modified environment while providing new information about the spatial ecology of this widespread predator in Europe.

Accelerated Exploration of Empty Material Compositional Space: Mg-Fe-B Ternary Metal Borides.

Borides are a family of materials with valuable properties for various applications. Their diverse structures and compositions, yet disparity in the constituent chemical elements for the known compounds, give elemental substitutions for prototypes great potential for material discovery. To explore uncharted material compositional space, we develop a workflow that joins high-throughput crystal structure prediction and automated diffraction pattern matching to discover new compounds with significant prediction and synthesis hurdles. Utilizing the workflow, we explore the empty Mg-Fe-B ternary compositional space, previously uncharted largely due to the immiscibility of Mg and Fe, as a paradigm. A total of 275 ternary boride prototypes are classified, using which we predict 23 (158) stable and metastable ternary phases within 50 (200) meV/atom above the convex hull. We identify Gd2(FeB)7-type Mg2Fe7B7 and ZrCo3B2-type MgFe3B2 to match previously unsolved experimental powder X-ray diffraction (PXRD) patterns. The discovered Mg2Fe7B7 and related channeled structures feature mismatched Mg and (FeB) sublattice periods, for which we conduct structural analyses with respect to the PXRD. They are predicted to exhibit exceptionally fast superionic transport of Mg and outstanding electrochemical performance, which serve as post-Li-ion battery candidate electrode materials. This result opens a new avenue for borides' potential applications as electrode materials and fast ionic conductors. This work also portrays the map and landscape of ternary metal borides with similar chemical environments. With high efficiency, the prototype- and PXRD-assisted crystal structure prediction workflow opens a new avenue for exploring various material compositional spaces across the periodic table.

Research on nonlinear second-order multi-agent systems containment control method with inherent time-delay based on compensator

In this study, a nonlinear second-order multi-agent systems containment control method based on compensator is discussed. In order to deal with nonlinear systems with inherent time delay, a new compensator-based control algorithm is proposed. The fully distributed compensator is designed to effectively converge the states of all followers to the convex hull formed by the states of leaders. By introducing a compensator, the errors in the system can be corrected and the influence of time delay on the system can be compensated to ensure that the system can achieve the desired performance and stability. Then, the original system is transformed into an error system by using matrix theory, and the stability of the error system is analyzed by using Lyapunov method, and the sufficient conditions for the solvability of the control problem are obtained. Finally, the effectiveness of the control algorithm is verified by matlab simulation.

Totally Symmetric Self-Complementary Plane Partition Matrices and Related Polytopes

AbstractPlane partitions in the totally symmetric self-complementary symmetry class (TSSCPPs) are known to be equinumerous with $$n\times n$$ n × n alternating sign matrices, but no explicit bijection is known. In this paper, we give a bijection from these plane partitions to $$\{0,1,-1\}$$ { 0 , 1 , - 1 } -matrices we call magog matrices, some of which are alternating sign matrices. We explore enumerative properties of these matrices related to natural statistics such as inversion number and number of negative ones. We then investigate the polytope defined as their convex hull. We show that all the magog matrices are extreme and give a partial inequality description. Finally, we define another TSSCPP polytope as the convex hull of TSSCPP boolean triangles and determine its dimension, inequalities, vertices, and facets.

Performance of an AI-powered visualization software platform for precision surgery in breast cancer patients

Surgery remains the primary treatment modality in the management of early-stage invasive breast cancer. Artificial intelligence (AI)-powered visualization platforms offer the compelling potential to aid surgeons in evaluating the tumor’s location and morphology within the breast and accordingly optimize their surgical approach. We sought to validate an AI platform that employs dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) to render three-dimensional (3D) representations of the tumor and 5 additional chest tissues, offering clear visualizations as well as functionalities for quantifying tumor morphology, tumor-to-landmark structure distances, excision volumes, and approximate surgical margins. This retrospective study assessed the visualization platform’s performance on 100 cases with ground-truth labels vetted by 2 breast-specialized radiologists. We assessed features including automatic AI-generated clinical metrics (e.g., tumor dimensions) as well as visualization tools including convex hulls at desired margins around the tumor to help visualize lumpectomy volume. The statistical performance of the platform’s automated features was robust and within the range of inter-radiologist variability. These detailed 3D tumor and surrounding multi-tissue depictions offer both qualitative and quantitative comprehension of cancer topology and may aid in formulating an optimal surgical approach for breast cancer treatment. We further establish the framework for broader data integration into the platform to enhance precision cancer care.

On containment control for discrete-time switching networks of delayed first-order systems

This paper delves into the domain of dynamic containment control, with specific focus on discrete-time perturbed and delayed systems characterised by first-order dynamics. Central to containment control is the aspiration to guide agents towards desired spatial formations while mitigating the effects of disturbances, ensuring cohesive behaviour. In this respect, we propose a containment protocol that ensures input-to-state stability with respect to the convex hull generated by the leader trajectories while guaranteeing robustness with respect to both fixed and distributed communication delays acting on the state measurements exchanged between the network nodes. Numerical simulations on a team of interconnected drones show the effectiveness of the proposed approach.

Extended formulations for binary optimal control problems

AbstractExtended formulations are an important tool in polyhedral combinatorics. Many combinatorial optimization problems require an exponential number of inequalities when modeled as a linear program in the natural space of variables. However, by adding artificial variables, one can often find a small linear formulation, i.e., one containing a polynomial number of variables and constraints, such that the projection to the original space of variables yields a perfect linear formulation. Motivated by binary optimal control problems with switching constraints, we show that a similar approach can be useful also for optimization problems in function space, in order to model the closed convex hull of feasible controls in a compact way. More specifically, we present small extended formulations for switches with bounded variation and for dwell-time constraints. For general linear switching point constraints, we devise an extended model linearizing the problem, but show that a small extended formulation that is compatible with discretization cannot exist unless P = NP.

The limit as $$s\nearrow 1$$ of the fractional convex envelope

The limit as $$s\nearrow 1$$ of the fractional convex envelope