We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point is necessarily found when a limit point satisfies Robinson's condition. We conduct numerical experiments minimizing nonlinear functions subject to a copositive cone constraint. In order to do this, we consider a well known polyhedral approximation of this cone by means of refining the polyhedral constraints after each augmented Lagrangian iteration. In some numerical experiments, our strategy outperforms the standard approach of considering a close polyhedral approximation of the full copositive cone in every iteration.

General Polyhedral Approximation of two-stage robust linear programming for budgeted uncertainty

<p><strong>Context and relevance.</strong> The time-optimization problem is solved for a linear stationary system with discrete-time and summary first-order constraints on control. <strong>Objective.</strong> Demonstrate the possibility of constructing the guaranteeing control in the time-optimization problem. <strong>Hypothesis. </strong>The guaranteed solution found by applying the polyhedral approximation algorithm will converge to the optimal one. <strong>Methods and materials.</strong> This task has a number of features that complicate its solution using standard methods such as dynamic programming and the maximum principle. For this reason, it is proposed to use a geometric approach and an apparatus of null-controllable sets to solve the time-optimization problem. <strong>Results.</strong> For the case when the summary constraints are piecewise-linear, it is proved that all null-controllable sets are polyhedrons, which makes it possible to reduce the initial control problem to solving a number of linear programming problems. For arbitrary convex summary constraints, the possibility of constructing a guaranteeing solution in terms of time-optimization problem based on polyhedral approximation methods is shown.<strong> Conclusions.</strong> The convergence of the guaranteeing solution to the optimal one is investigated: it is proved that it will be completed in a finite number of iterations if the polyhedral approximation algorithm used guarantees convergence in the sense of the Hausdorff metric. The developed theoretical results are demonstrated using a numerical example.</p>

Recovering the polyhedral geometry of fragments.

Nonconvex quadratically constrained programs (QCPs) are generally NP-hard and challenging problems. In this paper, we propose two novel mixed-integer linear programming (MILP) approximations for a nonconvex QCP. Our method begins by utilizing an eigenvalue-based decomposition to express the nonconvex quadratic function as the difference of two convex functions. We then introduce an additional variable to partition each nonconvex constraint into a second-order cone (SOC) constraint and the complement of an SOC constraint. We employ two polyhedral approximation approaches to approximate the SOC constraint. The complement of an SOC constraint is approximated using a combination of linear and complementarity constraints. As a result, we approximate the nonconvex QCP with two linear programs with complementarity constraints (LPCCs). More importantly, we prove that the optimal values of the LPCCs asymptotically converge to that of the original nonconvex QCP. By proving the boundedness of the LPCCs, we further reformulate the LPCCs as MILPs. We demonstrate the effectiveness of our approaches via numerical experiments by applying our proposed approximations to randomly generated instances and two application problems: the joint decision and estimation problem and the two-trust-region subproblem. The numerical results show significant advantages of our approaches in terms of solution quality and computational time compared with existing benchmark approaches. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods and Analysis. Funding: K. Pan was supported in part by the Research Grants Council of Hong Kong [Grant 15503723]. J. Cheng and B. Yang were supported in part by the Office of Naval Research [Grant N00014-20-1-2154]. J. Cheng was supported in part by the National Science Foundation [Grant ECCS-2404412]. B. Yang was supported in part by the Air Force Office of Scientific Research [Grant FA9550-23-1-0508]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0719 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0719 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A Progressive Polyhedral Approximation Method for Nonlinear PDE-Constrained Electricity-Water Nexus Dispatch

We introduce a generic technique to obtain linear relaxations of semidefinite programs with provable guarantees based on the commutativity of the constraint and the objective matrices. We study conditions under which the optimal value of the SDP and the proposed linear relaxation match, which we then relax to provide a flexible methodology to derive effective linear relaxations. We specialize these results to provide linear programs that approximate well-known semidefinite programs for the max cut problem proposed by Poljak and Rendl, and the Lovász theta number; we prove that the linear program proposed for max cut certifies a known eigenvalue bound for the maximum cut value and is in fact stronger. Our ideas can be used to warm-start algorithms that solve semidefinite programs by iterative polyhedral approximation of the feasible region. We verify this capability through multiple experiments on the max cut semidefinite program, the Lovász theta number and on three families of semidefinite programs obtained as convex relaxations of certain quadratically constrained quadratic problems.

The article discusses the development of software for solving the speed-in-action problem for linear systems with discrete time and first-order constraints (geometric or summary). Algorithms for constructing an optimal process are formulated and the main theoretical information necessary for their derivation is given. In particular, the discussed solution method is based on the apparatus of null-controllable sets, which was developed in previous theoretical works, and the dynamic programming method. The computational complexity of these algorithms is estimated. It is demonstrated that the solution of the speed-in-action problem can be performed in polynomial time depending on the minimum time (in the case of geometric constraints, this requires additional polyhedral approximations of null-controllable sets). A description of the computer program, all its structural elements and interface, composed using the programming language MatLab, is presented. The program uses the Quick Hull algorithm as its main tool when constructing null-controllable sets, and the interior point method for solving linear and quadratic programming problems when calculating optimal control and polyhedral approximation, respectively. The program output is provided in graphical and text formats. The operation of the software is demonstrated by the example of solving the problem of the fastest stabilization of glucose and insulin levels in blood plasma. Two mathematical models are considered: with restrictions on the volume of a singe injection of insulin and glucose and with restrictions on the total permissible volume of all injections. The first statement corresponds to geometric constraints on control, the second statement is described by summary constraints.

A polyhedral approximation of the yield domain for masonry sections

ABSTRACT This paper examines the issue of collision‐avoidance trajectory tracking control in a multi‐quadrotor unmanned aerial vehicle (UAV) slung load system, with particular emphasis on the scenario where the reference trajectory is unreachable. The challenge of tracking an unreachable reference trajectory is effectively addressed by integrating a trajectory planner and a trajectory tracking controller within a unified distributed model predictive control (DMPC) framework. Moreover, the nonlinear system is linearized using the first‐order Taylor approximation, significantly simplifying the computation in DMPC. To ensure collision avoidance with both dynamic and static obstacles, the MINVO basis is employed to calculate the minimum volume of the exterior polyhedral approximation of the obstacles' trajectories, which is significantly smaller than that achieved using the B‐spline or Bernstein bases typically utilized in the planning literature. Simulation experiments involving four UAVs, one payload, two static obstacles, and one dynamic obstacle are conducted to evaluate the effectiveness of the proposed DMPC method.

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.

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 intersection of an affine subspace with the cone of positive semidefinite matrices is called a spectrahedron. An orthogonal projection thereof is called a spectrahedral shadow or projected spectrahedron. Spectrahedra and their projections can be seen as a generalization of polyhedra. This article is concerned with the problem of approximating the recession cones of spectrahedra and spectrahedral shadows via polyhedral cones. We present two iterative algorithms to compute outer and inner approximations to within an arbitrary prescribed accuracy. The first algorithm is tailored to spectrahedra and is derived from polyhedral approximation algorithms for compact convex sets and relies on the fact, that an algebraic description of the recession cone is available. The second algorithm is designed for projected spectrahedra and does not require an algebraic description of the recession cone, which is in general more difficult to obtain. We prove correctness and finiteness of both algorithms and provide numerical examples.

Abstract The 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.

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.

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.

<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>

We prove that any length metric space homeomorphic to a 2manifold with boundary, also called a length surface, is the Gromov-Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical uniformization theorem for Riemann surfaces and a limiting argument, we establish a general "one-sided" quasiconformal uniformization theorem for length surfaces with locally finite Hausdorff 2-measure. Our approach yields a new proof of the Bonk-Kleiner theorem characterizing Ahlfors 2-regular quasispheres.

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.