Non-convex Research Articles

Three-Dimensional Flight Corridor: An Occupancy Checking Process for Unmanned Aerial Vehicle Motion Planning inside Confined Spaces

To deploy Unmanned Aerial Vehicles (UAVs) inside heterogeneous GPS-denied confined (potentially unknown) spaces, such as those encountered in mining and Urban Search and Rescue (USAR), requires the enhancement of numerous technologies. Of special interest is for UAVs to identify collision-freeSafe Flight Corridors (SFC+) within highly cluttered convex- and non-convex-shaped environments, which requires UAVs to perform advanced flight maneuvers while exploiting their flying capabilities. Within this paper, a novel auxiliary occupancy checking process that augments traditional 3D flight corridor generation is proposed. The 3D flight corridor is established as a topological structure based on a hand-crafted path either derived from a computer-generated environment or provided by the human operator, which captures humans’ preferences and desired flight intentions for the given space. This corridor is formulated as a series of interconnected overlapping convex polyhedra bounded by the perceived environmental geometries, which facilitates the generation of suitable 3D flight paths/trajectories that avoid local minima within the corridor boundaries. An occupancy check algorithm is employed to reduce the search space needed to identify 3D obstacle-free spaces in which their constructed polyhedron geometries are replaced with alternate convex polyhedra. To assess the feasibility and efficiency of the proposed SFC+ methodology, a comparative study is conducted against the Star-Convex Method (SCM), a prominent algorithm in the field. The results reveal the superiority of the proposed SFC+ methodology in terms of its computational efficiency and reduced search space for UAV maneuvering solutions. Various challenging confined-environment scenarios, each with different obstacle densities (confined scenarios), are utilized to verify the obtained outcomes.

A single-stage optimization procedure for data envelopment analysis

In data envelopment analysis, a conventional procedure for testing efficiency of decision making units (DMUs) consists of two stages, each requiring solution of a linear program. The first stage identifies the input or output radial efficiency of a DMU, and the second stage maximizes the sum of component input and output slacks. A traditional alternative is the single-stage approximation of the two-stage procedure in which the objective functions of the first and second stages are combined, with the latter multiplied by a small positive constant epsilon. A known drawback of such approach is that very small values of epsilon may cause computational inaccuracies and large values do not allow a good approximation. In this paper, we develop a new single-stage linear programming approach that does not require the specification of a small constant epsilon and is completely equivalent to the conventional two-stage solution procedure. It produces exactly the same sets of primal and dual optimal solutions as the two separate stages of the two-stage approach. The new single-stage procedure is applicable to any convex polyhedral technology.

Cyber-Physical Interdependent Restoration Scheduling for Active Distribution Network via Ad Hoc Wireless Communication

This paper proposes a post-disaster cyber-physical interdependent restoration scheduling (CPIRS) framework for active distribution networks (ADN) where the simultaneous damages on cyber and physical networks are considered. The ad hoc wireless device-to-device (D2D) communication is leveraged, for the first time, to establish cyber networks instantly after the disaster to support ADN restoration. The repair and operation crew dispatching, the remote-controlled network reconfiguration and the system operation with DERs can be effectively coordinated under the cyber-physical interactions. The uncertain outputs of renewable energy resources (RESs) are represented by budget-constrained polyhedral uncertainty sets. Through implementing linearization techniques on disjunctive expressions, a monolithic mixed-integer linear programming (MILP) based two-stage robust optimization model is formulated and subsequently solved by a customized column-and-constraint generation (C&CG) algorithm. Numerical results on the IEEE 123-node distribution system demonstrate the effectiveness and superiorities of the proposed CPIRS method for ADN.

Reference tracking via output feedback for constrained uncertain linear systems

In this paper we propose a robust output feedback control scheme for constant reference tracking in uncertain linear systems subject to state and control constraints. First, we establish conditions under which a polyhedral set is Output Feedback Controlled Invariant with respect to a linear system subject to polytopic uncertainties, i.e. a set guaranteeing robust constraint satisfaction via output feedback. Then, we build a dynamic output feedback controller for the uncertain model based on set-membership observers, allowing the reduction of the set of possible states consistent with the measurements. Finally, to reduce the tracking error, we propose a model update procedure that adjusts the nominal model used for tracking to the output measurements. Numerical experiments illustrate the ability of the proposed controller to achieve reference tracking with reduced offset for the uncertain systems under consideration. To the best of our knowledge, it is one of the first works that addresses constant reference tracking in uncertain linear systems under constraints with output feedback.

Asynchronous secure control for singular nonhomogeneous Markov jump cyber‐physical systems against dual cyber‐attacks

AbstractIn this article, based on the dynamic event‐triggered transmission protocol, the issue of asynchronous secure control for singular nonhomogeneous Markov jump cyber‐physical systems (CPSs) is studied. The matrix that is governed by the time‐varying transition rates lies in a convex polyhedron. To further degrade the transmission percentage over the network, the dynamic event‐triggered transmission protocol is developed. Besides, owing to the vulnerability of the network, the replay attack and deception attack are both considered, the occurrences of which are modelled by mutual independent random variables satisfying Bernoulli distribution. By constructing a parameter dependent and an auxiliary dynamic variable dependent Lyapunov functional, sufficient conditions of stochastic admissible for singular nonhomogeneous Markov jump CPSs are derived by parameter dependent matrix inequalities, which can be converted into the finite linear matrix inequalities. A numerical example is provided to illustrate the feasibility and effectiveness of the theoretical results.

Algorithm for density of angle distribution in random sections of polyhedron

Scanning electron micrographs of solid materials are a source of analysis for hardness, defects, etc. A micrograph is a two-dimensional image of a material being cut. Typical photomicrographs contain about a hundred thousand structural elements, such as grain boundaries, boundary junctions, etc., which are not easy to analyze with the naked eye. Currently, computer vision can be used to analyze the geometric properties of structural elements. One of the important properties is the cross section of the grain, which is the result of cutting the material. In the article we turn to one of the geometric properties of random grain cuttings. The statistical properties of the cross sections of grains in an alloy are equivalent to the statistical properties of the cross sections of individual grains by random planes if the grains do not touch each other. We present two stages of mathematical analysis of the grain cross-sections. First, using the algorithm for generating random planes with a uniform distribution, a method is proposed for generating random sections of polyhedra. The method can be useful in stochastic geometry and computational geometry, as well as in materials science. Second, sections of polyhedra with random planes are considered, and a numerical algorithm for estimating the distribution density of angles of convex polyhedra is also implemented. Examples of distribution densities in random sections for a cube, a regular triangular prism, and a truncated prism are given. The developed algorithm and the obtained distribution densities are of practical use in the analysis of micrographs of thin sections of crystalline alloys.

The robust cyclic job shop problem

This paper deals with the cyclic job shop problem where the task durations are uncertain and belong to a polyhedral uncertainty set. We formulate the cyclic job shop problem as a two-stage robust optimization model. The cycle time and the execution order of tasks executed on the same machines correspond to the here-and-now decisions and have to be decided before the realization of the uncertainty. The starting times of tasks corresponding to the wait-and-see decisions are delayed and can be adjusted after the uncertain parameters are known. In the last decades, different solution approaches have been developed for two-stage robust optimization problems. Among them, the use of affine policies, row and row-and-column generation algorithms are the most common. In this paper, we propose a branch-and-bound algorithm to tackle the robust cyclic job shop problem with cycle time minimization. The algorithm uses, at each node of the search tree, a robust version of the Howard’s algorithm to derive a lower bound on the optimal cycle time. Moreover, we design a row generation algorithm and a column-and-row generation algorithm and compare it to the branch-and-bound method. Finally, encouraging preliminary results on numerical experiments performed on randomly generated instances are presented.

Weak discrete maximum principle of isoparametric finite element methods in curvilinear polyhedra

The weak maximum principle of the isoparametric finite element method is proved for the Poisson equation under the Dirichlet boundary condition in a (possibly concave) curvilinear polyhedral domain with edge openings smaller than π \pi , which include smooth domains and smooth deformations of convex polyhedra. The proof relies on the analysis of a dual elliptic problem with a discontinuous coefficient matrix arising from the isoparametric finite elements. Therefore, the standard H 2 H^2 elliptic regularity which is required in the proof of the weak maximum principle in the literature does not hold for this dual problem. To overcome this difficulty, we have decomposed the solution into a smooth part and a nonsmooth part, and estimated the two parts by H 2 H^2 and W 1 , p W^{1,p} estimates, respectively. As an application of the weak maximum principle, we have proved a maximum-norm best approximation property of the isoparametric finite element method for the Poisson equation in a curvilinear polyhedron. The proof contains non-trivial modifications of Schatz’s argument due to the nonconformity of the iso-parametric finite elements, which requires us to construct a globally smooth flow map which maps the curvilinear polyhedron to a perturbed larger domain on which we can establish the W 1 , ∞ W^{1,\infty } regularity estimate of the Poisson equation uniformly with respect to the perturbation.

Polyhedral estimation of L1 and L∞ incremental gains of nonlinear systems

We provide novel dissipativity conditions for bounding the incremental L1 gain of systems. Moreover, we adapt existing results on the L∞ gain to the incremental setting and relate the incremental L1 and L∞ gain bounds through transposed systems. Building on work on optimization based approaches to constructing polyhedral Lyapunov functions, we make use of these conditions to obtain a Linear Programming based algorithm that can provide increasingly sharp bounds on the gains as a function of a given candidate polyhedral storage function or polyhedral set. The algorithm is also extended to allow for the design of constrained linear feedback controllers for performance, as measured by the bounds on the incremental gains. We apply the algorithm to a couple of numerical examples to illustrate the power, as well as some limitations, of this approach.

Angle Sums of Random Polytopes

For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in Rd. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.

A State of the Art Review of Systems of Linear Inequalities and Related Observability Problems

This work is a short review of the state of the art aiming to contribute to the use, disclosure, and propagation of systems of linear inequalities in real life, teaching, and research. It shows that the algebraic structure of their solutions consists of the sum of a linear subspace, an acute cone, and a polytope, and that adequate software exists to obtain, in their simplest forms, these three components. The work describes, based on orthogonality and polarity, homogeneous and complete systems of inequalities, the associated compatibility problems, and their relations with convex polyhedra and polytopes, which are the only possible solution for bounded problems, the most common in real practice. The compatibility and the observability problems, including their symbolic forms, are analyzed and solved, identifying the subsets of unknowns with unique solutions and those unbounded, important items of information with practical relevance in artificial intelligence and automatic learning. Having infinitely many solutions of a given problem allows us to find solutions when some of the assumptions fail and unexpected constraints come into play, a common situation for engineers. The linear programming problem becomes trivial when the set of all solutions is available and all solutions are obtained, contrary to the case of standard programs that provide only one solution. Several examples of applications to several areas of knowledge are presented, illustrating the advantages of solving these systems of inequalities.

Complete Characterization of Polyhedral Self-Affine Tiles

A self-affine tile is a compact set Gsubset {mathbb R}^d that admits a partition (tiling) by parallel shifts of the set M^{-1}G, where M is an expanding matrix. We find all self-affine tiles which are polyhedral sets, i.e., unions of finitely many convex polyhedra. It is shown that there exists an infinite family of such polyhedral sets, not affinely equivalent to each other. A special attention is paid to integral self-affine tiles with standard digit sets, when the matrix M and the translation vectors are integer. Applications to the approximation theory and to the functional analysis are discussed.

Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

Distributed robust optimization with coupled constraints via Tseng’s splitting method

We consider distributed robust convex optimization with both local and coupled constraints, where coefficients with respect to local constraints are uncertain in polyhedral sets. We derive an equivalent optimization model with extended decision variables and finite constraints, and characterize its optimality condition. Also, we transform the problem as finding zeros of an operator and reveal the monotonicity of the underlying operator. By employing Tseng’s operator splitting method, we design a distributed algorithm to solve the considered robust optimization problem with guaranteed convergence, and also illustrate the efficiency in numerical simulations.

A branch-and-prune algorithm for discrete Nash equilibrium problems

AbstractWe present a branch-and-prune procedure for discrete Nash equilibrium problems with a convex description of each player’s strategy set. The derived pruning criterion does not require player convexity, but only strict convexity of some player’s objective function in a single variable. If satisfied, it prunes choices for this variable by stating activity of certain constraints. This results in a synchronous branching and pruning method. An algorithmic implementation and numerical tests are presented for randomly generated instances with convex polyhedral strategy sets and convex quadratic as well as non-convex quadratic objective functions.

An extension to Voro++ for multithreaded computation of Voronoi cells

Voro++ is a software library written in C++ for computing the Voronoi tessellation, a technique in computational geometry that is widely used for analyzing systems of particles. Voro++ was released in 2009 and is based on computing the Voronoi cell for each particle individually. Here, we take advantage of modern computer hardware, and extend the original serial version to allow for multithreaded computation of Voronoi cells via the OpenMP application programming interface. We test the performance of the code, and demonstrate that it can achieve parallel efficiencies greater than 95% in many cases. The multithreaded extension follows standard OpenMP programming paradigms, allowing it to be incorporated into other programs. We provide an example of this using the VoroTop software library, performing a multithreaded Voronoi cell topology analysis of up to 102.4 million particles. Program summaryProgram title:Voro++CPC Library link to program files:https://doi.org/10.17632/tddc4w4zkk.1Developer's repository link:https://github.com/chr1shr/voroLicensing provisions: BSD 3-clause (with LBNL modification)Programming language: C++External routines/libraries: OpenMPNature of problem: Multithreaded computation of the Voronoi tessellation in two and three dimensionsSolution method: The Voro++ library is built around several C++ classes that can be incorporated into other programs. The two largest components are the container... classes that spatially sort input particles into a grid-based data structure, allowing for efficient searches of nearby particles, and the voronoicell... classes that represent a single Voronoi cell as an arbitrary convex polygon or polyhedron. The Voronoi cell for each particle is built by considering a sequence of plane cuts based on neighboring particles, after which many different statistics (e.g. volume, centroid, number of vertices) can be computed. Since each Voronoi cell is calculated individually, the Voronoi cells can be computed using multithreading via OpenMP.

Covariance edges matrix of geometric elements

In this paper, we introduce a new matrix associated with polygons and polyhedrons, namely the covariance edges matrix. We show that, for a regular polygon or polyhedron the corresponding matrix is proportional to the identity of size two or three. Based on this fact, we propose, as an application, several algebraic shape quality measures for convex polygons or polyhedrons. Furthermore, this matrix may be related to the metric of a simplex. Future studies will be devoted to the definition of the covariance edges matrix for higher elements and real applications to mesh optimisation.

Convergence to Infinite-Dimensional Compound Poisson Distributions on Convex Polyhedra

Convergence to Infinite-Dimensional Compound Poisson Distributions on Convex Polyhedra

On the Complexity of PAC Learning in Hilbert Spaces

We study the problem of binary classification from the point of view of learning convex polyhedra in Hilbert spaces, to which one can reduce any binary classification problem. The problem of learning convex polyhedra in finite-dimensional spaces is sufficiently well studied in the literature. We generalize this problem to that in a Hilbert space and propose an algorithm for learning a polyhedron which correctly classifies at least 1 − ε of the distribution, with a probability of at least 1 − δ, where ε and δ are given parameters. Also, as a corollary, we improve some previous bounds for polyhedral classification in finite-dimensional spaces.

Parameter Identification of Bivariate Fractal Interpolation Surfaces by Using Convex Hulls

The scope of this article is to identify the parameters of bivariate fractal interpolation surfaces by using convex hulls as bounding volumes of appropriately chosen data points so that the resulting fractal (graph of) function provides a closer fit, with respect to some metric, to the original data points. In this way, when the parameters are appropriately chosen, one can approximate the shape of every rough surface. To achieve this, we first find the convex hull of each subset of data points in every subdomain of the original lattice, calculate the volume of each convex polyhedron and find the pairwise intersections between two convex polyhedra, i.e., the convex hull of the subdomain and the transformed one within this subdomain. Then, based on the proposed methodology for parameter identification, we minimise the symmetric difference between bounding volumes of an appropriately selected set of points. A methodology for constructing continuous fractal interpolation surfaces by using iterated function systems is also presented.