Polyhedral Approximation Research Articles

Constructing Efficient Mesh-Based Global Grid Systems with Reduced Distortions

Recent advancements in geospatial technologies have significantly expanded the volume and diversity of geospatial data, unlocking new and innovative applications that require novel Geographic Information Systems (GIS). (Discrete) Global Grid Systems (DGGSs) have emerged as a promising solution to further enhance modern geospatial capabilities. Current DGGSs employ a simple, low-resolution polyhedral approximation of the Earth for efficient operations, but require a projection between the Earth’s surface and the polyhedral faces. Equal-area DGGSs are desirable for their low distortion, but they fall short of this promise due to the inefficiency of equal-area projections. On the other hand, efficiency-first DGGSs need to better address distortion. We introduce a novel mesh-based DGGS (MBD) which generalizes efficient operations over watertight triangular meshes with spherical topology. Unlike traditional approaches that rely on Platonic or Catalan solids, our mesh-based method leverages high-resolution spherical meshes to offer greater flexibility and accuracy. MBD allows high-resolution polyhedra (HRP) to be used as the base polyhedron of a DGGS, significantly reducing distortion. To address the operational challenges, we introduce a new hash encoding method and an efficient barycentric indexing method (BIM). MBD extends Atlas of Connectivity Maps to the BIM to provide efficient spatial and hierarchical traversal. We introduce several new base polyhedra with lower areal and angular distortion, and we experimentally validate their properties and demonstrate their efficiency. Our experimentation shows that we achieve constant-time operations for high-resolution MBD, and we recommend polyhedra to be used as the base polyhedron for low-distortion DGGSs, compact faces, and efficient operations.

Polyhedral approximation and uniformization for non-length surfaces

We prove that any metric surface (that is, metric space homeomorphic to a 2 -manifold with boundary) with locally finite Hausdorff 2 -measure is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2 -sphere with finite Hausdorff 2 -measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala–Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk–Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain.

Dispatchable region for distributed renewable energy generation in reconfigurable AC–DC distribution networks with soft open points

The dispatchable region (DR) describes the ability of a power system to adapt to the variability of distributed renewable energy (DRE) generation. Switchable devices such as soft open points (SOPs) and tie switches enable the AC–DC distribution network to flexibly cope with DRE power fluctuations. Currently, research on DRs seldom considers the effects of SOPs and reconfiguration with tie switches in AC–DC distribution networks, which prevents accurate evaluation of DR for this type of network. To bridge this gap, this paper formulates DRs for DRE generation in an AC–DC distribution network considering SOPs and reconfiguration with tie switches. The original model for deriving DRs is typically nonconvex, and the DR boundaries are intractable because of nonlinear power flow equations and binary variables of tie switches. Accordingly, this paper employs the outer polyhedral approximation and convex hull relaxation to approximate the original DR. An efficient adaptive constraint generation algorithm is developed to search for boundaries of the approximated DR. The numerical results show that the proposed model can effectively adapt to the DR of reconfigurable AC–DC distribution networks with SOPs and verify the accuracy and computational efficiency of the proposed method.

A polyhedral approximation algorithm for recession cones of spectrahedral shadows

A polyhedral approximation algorithm for recession cones of spectrahedral shadows

Grid‐side flexibility assessment based on security distance of dispatchable region for AC/DC hybrid system using approximate second‐order cone relaxation

AbstractThe rapid growth of offshore wind power has resulted in a mismatch between generation and demand because of its variability. To quantify the maximum wind power penetration of the transmission network, the dispatchable region is defined as the largest region in the uncertainty space. The security distance is defined as the minimum distance from the operating point to the boundaries of the dispatchable region. System operators can use security distance as a metric to assess the flexibility of the power system. This paper proposes a method to construct the dispatchable region for the AC/DC hybrid system with VSC‐HVDC by outer convex relaxation firstly. The second‐order cone relaxation is employed to reformulate non‐convex and non‐linear power flow equations. Next, a polyhedral approximation is adopted to obtain the convex hull of the dispatchable region. Subsequently, an efficient algorithm known as adaptive constraint generation (Ad‐CG) is introduced to calculate the boundaries of the dispatchable region. Furthermore, solving the Chebyshev centre problem determines the minimal security distance. The modified IEEE 5‐bus and 39‐bus system is used for validating effectiveness of the proposed method and evaluating the impact of the converter reactive power compensation capacity and generation dispatch on the dispatchable region.

Polyhedral Approximation of Spectrahedral Shadows via Homogenization

This article is concerned with the problem of approximating a not necessarily bounded spectrahedral shadow, a certain convex set, by polyhedra. By identifying the set with its homogenization, the problem is reduced to the approximation of a closed convex cone. We introduce the notion of homogeneous δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-approximation of a convex set and show that it defines a meaningful concept in the sense that approximations converge to the original set if the approximation error δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} diminishes. Moreover, we show that a homogeneous δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-approximation of the polar of a convex set is immediately available from an approximation of the set itself under mild conditions. Finally, we present an algorithm for the computation of homogeneous δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-approximations of spectrahedral shadows and demonstrate it on examples.

Risk-Based Hosting Capacity Analysis in Distribution Systems

Solar hosting capacity analysis (HCA) assesses the ability of a distribution network to host distributed solar generation without seriously violating distribution network constraints. In this paper, we consider risk-sensitive HCA that limits the risk of network constraint violations with a collection of scenarios of solar irradiance and nodal power demands, where risk is modeled via the conditional value at risk (CVaR) measure. First, we consider the question of maximizing aggregate installed solar capacities, subject to risk constraints and solve it as a second-order cone program (SOCP) with a standard conic relaxation of the feasible set of the power flow equations. Second, we design an incremental algorithm to decide whether a configuration of solar installations has acceptable risk of constraint violations, modeled via CVaR. The algorithm circumvents explicit risk computation by incrementally constructing inner and outer polyhedral approximations of the set of acceptable solar installation configurations from prior such tests conducted. Our numerical examples study the impact of risk parameters, the number of scenarios and the scalability of our framework.

Методы численного моделирования множеств 0-управляемости линейной дискретной динамической системы с ограниченным управлением на основе алгоритмов полиэдральной аппроксимации

<p>The article deals with the problem of constructing a polyhedral approximation of the 0-controllable sets of a linear discrete-time system with linear control constraints. To carry out the approximation, it is proposed to use two heuristic algorithms aimed at reducing the number of vertices of an arbitrary polyhedron while maintaining the accuracy of the description in the sense of the Hausdorff distance. The reduction of the problem of calculating the distance between nested polyhedra to the problem of convex programming is demonstrated. The issues of optimality of obtained approximations are investigated. Examples are given.</p>

Polyhedral approximation of metric surfaces and applications to uniformization

Polyhedral approximation of metric surfaces and applications to uniformization

Collision-Free Model Predictive Trajectory Tracking Control for UAVs in Obstacle Environment

In this paper, we propose a collision-free model predictive trajectory tracking control algorithm for unmanned aerial vehicles (UAVs) in environments with both static obstacles and dynamic obstacles. Collision avoidance is ensured by obtaining outer polyhedral approximations of each interval of the dynamic obstacles trajectories based on MINVO basis, and then optimizing a plane to separate the polyhedra and the trajectory of the UAV. By incorporating the resulting computationally efficient collisionfree constraints and divers physical constraints, a model predictive control (MPC) optimization problem is formulated with a tailored terminal constraint set, which can be solved by a standard nonlinear programming solver. Moreover, the control theoretic properties are established, including recursive feasibility, the guarantee of collision avoidance, as well as closed-loop stability. Finally, the efficacy of the proposed algorithm is successfully evaluated by a simulation in a multi-obstacle environment.

Elements of higher homotopy groups undetectable by polyhedral approximation

When non-trivial local structures are present in a topological space $X$, a common approach to characterizing the isomorphism type of the $n$-th homotopy group $\pi_n(X,x_0)$ is to consider the image of $\pi_n(X,x_0)$ in the $n$-th \v{C}ech homotopy group $\check{\pi}_n(X,x_0)$ under the canonical homomorphism $\Psi_{n}:\pi_n(X,x_0)\to \check{\pi}_n(X,x_0)$. The subgroup $\ker(\Psi_n)$ is the obstruction to this tactic as it consists of precisely those elements of $\pi_n(X,x_0)$, which cannot be detected by polyhedral approximations to $X$. In this paper, we use higher dimensional analogues of Spanier groups to characterize $\ker(\Psi_n)$. In particular, we prove that if $X$ is paracompact, Hausdorff, and $LC^{n-1}$, then $\ker(\Psi_n)$ is equal to the $n$-th Spanier group of $X$. We also use the perspective of higher Spanier groups to generalize a theorem of Kozlowski-Segal, which gives conditions ensuring that $\Psi_{n}$ is an isomorphism.

Virtual element method for elliptic bulk‐surface PDEs in three space dimensions

AbstractIn this work we present a novel bulk‐surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk‐surface partial differential equations in three space dimensions. The BSVEM is based on the discretization of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. We present a geometric error analysis of bulk‐surface polyhedral meshes independent of the numerical method. Then, we show that BSVEM has optimal second‐order convergence in space, provided the exact solution is in the bulk and on the surface, where the additional is due to the combined effect of surface curvature and polyhedral elements close to the boundary. We show that general polyhedra can be exploited to reduce the computational time of the matrix assembly. Two numerical examples on the unit sphere and on the Dupin ring cyclide confirm the convergence result.

Geometric Duality Results and Approximation Algorithms for Convex Vector Optimization Problems

We study geometric duality for convex vector optimization problems. For a primal problem with a -dimensional objective space, we formulate a dual problem with a -dimensional objective space. Consequently, different from an existing approach, the geometric dual problem does not depend on a fixed direction parameter, and the resulting dual image is a convex cone. We prove a one-to-one correspondence between certain faces of the primal and dual images. In addition, we show that a polyhedral approximation for one image gives rise to a polyhedral approximation for the other. Based on this, we propose a geometric dual algorithm which solves the primal and dual problems simultaneously and is free of direction-biasedness. We also modify an existing direction-free primal algorithm in such a way that it solves the dual problem as well. We test the performance of the algorithms for randomly generated problem instances by using the so-called primal error and hypervolume indicator as performance measures.

E-Carsharing siting and sizing DLMP-based under demand uncertainty

Electric vehicle (EV) sales and shared mobility are increasing worldwide. Despite its challenges, e-carsharing has an opportunity to still profit in periods of low rental demand compared to traditional carsharing. The purpose of this paper is to assess the profitability of an e–carsharing company based on distribution local marginal price (DLMP) and vehicle-to-grid (V2G) that cooperates with the distribution system operator (DSO) through a two-stage stochastic model. The AC optimal power flow (ACOPF) is modeled using second-order cone programming (SOCP) linearized by the global polyhedral approximation. The IEEE 33 bus test system and a real Kernel distribution for the EV rental demands are used in four planning cases in the GAMS environment. The results indicate that the proposed methodology does not affect EV user satisfaction. Moreover, the planning disregarding the power grid perspective is the most profitable, but the operation may not be possible in real applications due to the high-power flows via V2G. Finally, the e–carsharing planning considering the DSO perspective increased the charging cost by 1.66 % but also reduced the DLMP peak, losses, and peak demand by 2.5 %, 1.5 %, and 5.1 %, respectively. One important conclusion is that the technical benefits brought to the DSO by the e–carsharing company could be turned into services and advantages for both agents, increasing profit and mitigating negative impacts, such as higher operational costs.

{IBNC}: Integrated Boundary and Natural {CSG} for Polyhedra (Review, Simplifications, and Integration of Prior Art)

To improve rendering performance or reduce implementation difficulty, when rendering, animating, or processing a solid with curved faces, its polyhedral approximation is often used. The polyhedron, R , and each one of its faces are usually modeled in Brep (i.e. Boundary representation). To improve numeric accuracy and processing performance, it may be useful to also have an NCrep (i.e. Natural Constructive representation) that models R as the regularized CSG combination of the natural halfspaces, which are bounded by planes that contain the faces referenced in the Brep. We review several BtoNC (Brep to NCrep) conversion schemes and propose a simplification, a reformulation, and a novel combination of some of them to define a new NCrep representation, which we named MAS (for Match Aggregate of Sectors), where each cell (i.e. face, edge, vertex), c , of R is associated with a sector, an NCrep of the localization of R to the neighborhood of c . R , each face of R , and the sector of each vertex of R are the regularization of their MAS or of its complement. The proposed IBNC (Integrated Boundary and Natural CSG) annotates the Brep corners with easily computed binary tags, making it possible to process each MAS directly (i.e. without BtoNC conversion). • Integrated Boundary and CSG Representation for Polygons and Polyhedra. • Boundary-to-CSG Conversion for Polygons and Polyhedra. • Trimming CSG models of faces of a polyhedron from its Brep.

Reachable set estimation for spacecraft relative motion based on bang-bang principle

For spacecraft formation flight, the information of relative motion reachable set is very important, which can be used to predict the operating boundary of adjacent spacecraft and thus to ensure the safety of spacecraft operation. In this paper, we aim at developing a numerical method to approximate the reachable set for spacecraft relative motion. In particular, we focus on the quality of the approximation and the computational cost. Based on the bang-bang control principle, a polyhedral approximation algorithm is proposed to compute the reachable set of a relative motion spacecraft system. An inner approximation and an outer approximation of the reachable set for the system can be obtained. We prove that the approximation quality measured in Hausdorff distance can be guaranteed. The method is easy to implement and has low computational cost. Finally, the effectiveness of the algorithm is demonstrated by experimental simulation.

Corrected Curvature Measures

This paper proposes a new mathematical and computational tool for inferring the geometry of shapes known only through approximations such as triangulated or digital surfaces. The main idea is to decouple the position of the shape boundary from its normal vector field. To do so, we extend a classical tool of geometric measure theory, the normal cycle, so that it takes as input not only a surface but also a normal vector field. We formalize it as a current in the oriented Grassmann bundle \(\mathbb {R}^3 \times \mathbb {S}^2\). By choosing adequate differential forms, we define geometric measures like area, mean and Gaussian curvatures. We then show the stability of these measures when both position and normal input data are approximations of the underlying continuous shape. As a byproduct, our tool is able to correctly estimate curvatures over polyhedral approximations of shapes with explicit bounds, even when their natural normal are not correct, as long as an external convergent normal vector field is provided. Finally, the accuracy, convergence and stability under noise perturbation is evaluated experimentally onto digital surfaces.

A Norm Minimization-Based Convex Vector Optimization Algorithm

We propose an algorithm to generate inner and outer polyhedral approximations to the upper image of a bounded convex vector optimization problem. It is an outer approximation algorithm and is based on solving norm-minimizing scalarizations. Unlike Pascoletti–Serafini scalarization used in the literature for similar purposes, it does not involve a direction parameter. Therefore, the algorithm is free of direction-biasedness. We also propose a modification of the algorithm by introducing a suitable compact subset of the upper image, which helps in proving for the first time the finiteness of an algorithm for convex vector optimization. The computational performance of the algorithms is illustrated using some of the benchmark test problems, which shows promising results in comparison to a similar algorithm that is based on Pascoletti–Serafini scalarization.

On approximations of the PSD cone by a polynomial number of smaller-sized PSD cones

We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: “how closely can we approximate the set of unit-trace \(n \times n\) PSD matrices, denoted by D, using at most N number of \(k \times k\) PSD constraints?” In this paper, we prove lower bounds on N to achieve a good approximation of D by considering two constructions of an approximating set. First, we consider the unit-trace \(n \times n\) symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in \({\mathbb {R}}^n\). We prove that if this set is a good approximation of D, then the number of subspaces must be at least exponentially large in n for any \(k = o(n)\). Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial \({\varvec{S}}_+^k\)-extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have \({\varvec{S}}_+^k\)-extension complexity at least \(\exp ( C \cdot \min \{ \sqrt{n}, n/k \})\) where C is some absolute constant. In addition, we show that any set S such that \(D \subseteq S\) and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have \({\varvec{S}}_+^k\)-extension complexity at least \(\exp ( C \cdot \min \{ n^{1/3}, \sqrt{n/k} \})\). These results imply that the cone of \(n \times n\) PSD matrices cannot be approximated by a polynomial number of \(k \times k\) PSD constraints for any \(k = o(n / \log ^2 n)\). These results generalize the recent work of Fawzi (Math Oper Res 46(4):1479–1489, 2021) on the hardness of polyhedral approximations of \({\varvec{S}}_+^n\), which corresponds to the special case with \(k=1\).

On the Approximation of Unbounded Convex Sets by Polyhedra

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the connections between existing results before introducing a new notion of polyhedral approximation called left( varepsilon , delta right) -approximation that integrates the unbounded case in a meaningful way. Some basic results about left( varepsilon , delta right) -approximations are proved for general convex sets. In the last section, an algorithm for the computation of left( varepsilon , delta right) -approximations of spectrahedra is presented. Correctness and finiteness of the algorithm are proved.