Set Estimation Under Biconvexity Restrictions

A set in the Euclidean plane is said to be biconvex if, for some angle $\theta\in[0,\pi/2)$, all its sections along straight lines with inclination angles $\theta$ and $\theta+\pi/2$ are convex sets (i.e, empty sets or segments). Biconvexity is a natural notion with some useful applications in optimization theory. It has also be independently used, under the name of "rectilinear convexity", in computational geometry. We are concerned here with the problem of asymptotically reconstructing (or estimating) a biconvex set $S$ from a random sample of points drawn on $S$. By analogy with the classical convex case, one would like to define the "biconvex hull" of the sample points as a natural estimator for $S$. However, as previously pointed out by several authors, the notion of "hull" for a given set $A$ (understood as the "minimal" set including $A$ and having the required property) has no obvious, useful translation to the biconvex case. This is in sharp contrast with the well-known elementary definition of convex hull. Thus, we have selected the most commonly accepted notion of "biconvex hull" (often called "rectilinear convex hull"): we first provide additional motivations for this definition, proving some useful relations with other convexity-related notions. Then, we prove some results concerning the consistent approximation of a biconvex set $S$ and and the corresponding biconvex hull. An analogous result is also provided for the boundaries. A method to approximate, from a sample of points on $S$, the biconvexity angle $\theta$ is also given

Blaschke, Separation Theorems and some Topological Properties for Orthogonally Convex Sets

In this paper, we deal with analytic and geometric properties of orthogonally convex sets. We establish a Blaschke-type theorem for path-connected and orthogonally convex sets in the plane using orthogonally convex paths. The separation of these sets is established using suitable grids. Consequently, a closed and orthogonally convex set is represented by the intersection of staircase-halfplanes in the plane. Some topological properties of orthogonally convex sets in dimensional spaces are also given.Comment: 17 pages, 10 figures, adding more reference

Geometric-based Optimization Algorithms for Cable Routing and Branching in Cluttered Environments

The need for designing lighter and more compact systems often leaves limited space for planning routes for the connectors that enable interactions among the system’s components. Finding optimal routes for these connectors in a densely populated environment left behind at the detail design stage has been a challenging problem for decades. A variety of deterministic as well as heuristic methods has been developed to address different instances of this problem. While the focus of the deterministic methods is primarily on the optimality of the final solution, the heuristics offer acceptable solutions, especially for such problems, in a reasonable amount of time without guaranteeing to find optimal solutions. This study is an attempt to furthering the efforts in deterministic optimization methods to tackle the routing problem in two and three dimensions by focusing on the optimality of final solutions. The objective of this research is twofold. First, a mathematical framework is proposed for the optimization of the layout of wiring connectors in planar cluttered environments. The problem looks at finding the optimal tree network that spans multiple components to be connected with the aim of minimizing the overall length of the connectors while maximizing their common length (for maintainability and traceability of connectors). The optimization problem is formulated as a bi-objective problem and two solution methods are proposed: (1) to solve for the optimal locations of a known number of breakouts (where the connectors branch out) using mixed-binary optimization and visibility notion and (2) to find the minimum length tree that spans multiple components of the system and generates the optimal layout using the previously-developed convex hull based routing. The computational performance of these methods in solving a variety of problems is further evaluated. Second, the problem of finding the shortest route connecting two given nodes in a 3D cluttered environment is considered and addressed through deterministically generating a graphical representation of the collision-free space and searching for the shortest path on the found graph. The method is tested on sample workspaces with scattered convex polyhedra and its computational performance is evaluated. The work demonstrates the NP-hardness aspect of the problem which becomes quickly intractable as added components or increase in facets are considered

On hub location problems in geographically flexible networks

The authors were partially supported by research groups SEJ-584 and FQM-331 (Junta de Andalucia) and projects MTM2016-74983-C02-01 (Spanish Ministry of Education and Science/FEDER), FEDER-US-1256951, P18-FR-1422, P18-FR-2369 (Junta de Andalucia), CEI-3FQM331 (Andalucia Tech), and NetmeetData (Fundacion BBVA - Big Data 2019). We also would like to acknowledge Elena Fernandez (Universidad de Cadiz) for her useful and detailed comments on previous versions of this manuscript.In this paper, we propose an extension of the uncapacitated hub location problem where the potential positions of the hubs are not fixed in advance. Instead, they are allowed to belong to a region around an initial discrete set of nodes. We give a general framework in which the collection, transportation, and distribution costs are based on norm-based distances and the hub-activation setup costs depend not only on the location of the hub that are opened but also on the size of the region where they are placed. Two alternative mathematical programming formulations are proposed. The first one is a compact formulation while the second one involves a family of constraints of exponential size that we separate efficiently giving rise to a branch-and-cut algorithm. The results of an extensive computational experience are reported showing the advantages of each of the approaches.Junta de Andalucia SEJ-584 FQM-331 FEDER-US-1256951 P18-FR-1422 P18-FR-2369Spanish Government European Commission MTM2016-74983-C02-01Andalucia Tech CEI-3FQM331NetmeetData (Fundacion BBVA - Big Data 2019

Implementation of an Extended Kalman Filter Using Inertial Sensor Data for UAVs During GPS Denied Applications

Unmanned Aerial Vehicles (UAVs) are widely used across the industry and have a strong military application for defense. As UAVs become more accessible so does the increase of their applications, now being more limited by one’s imagination as opposed to the past where micro electric components were the limiting factor. Almost all of the applications require GPS or radio guidance. For more covert and longer range missions relying solely on GPS and radio is insufficient as the Unmanned Aerial System is vulnerable to malicious encounters like GPS Jamming and GPS Spoofing. For long range mission GPS denied environments are common where loss of signal is experienced. For autonomous flight GPS is a fundamental requirement. In this work an advanced inertial navigation system is proposed along with a programmable Pixhawk flight controller and Cube Black autopilot. A Raspberry Pi serves as a companion computer running autonomous flight missions and providing data acquisition. The advancement in inertial navigation comes from the implementation of a high end Analog Devices’ IMU providing input to an Extended Kalman Filter (EKF) to reduce error associated with measurement noise. The EKF is a efficient recursive computation applying the least-squares method. UAS flight controller simulations and calibrations were conducted to ensure the expected flight capabilities were achieved. The developed software and hardware was implemented in a Quadcopter build to perform flight test. Flight test data were used to analyze the performance post flight. Later, simulated feedback of the inertial navigation based state estimates (from flight test data) is performed to ensure reliable position data during GPS denied flight. The EKF applied to perform strapdown navigation was a limited success at estimating the vehicles’ inertial states but only when tuned for the specific flight trajectory. The predicted position was successfully converted to GPS data and passed to the autopilot in a LINUX based simulations ensuring autonomous mission capability is maintainable in GPS denied environments. The results from this research can be applied with ease to any vehicle operating with a Pixhawk controller and a companion computer of the appropriate processing capability

Optimally fast incremental Manhattan plane embedding and planar tight span construction

We describe a data structure, a rectangular complex, that can be used to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n x n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O(n^2), and to add a single point to a planar tight span in time O(n). As an application of this construction, we show how to test whether a given finite metric space embeds isometrically into the Manhattan plane in time O(n^2), and add a single point to the space and re-test whether it has such an embedding in time O(n).Comment: 39 pages, 15 figure

Sparse Regression via Range Counting

The sparse regression problem, also known as best subset selection problem, can be cast as follows: Given a set S of n points in ?^d, a point y? ?^d, and an integer 2 ? k ? d, find an affine combination of at most k points of S that is nearest to y. We describe a O(n^{k-1} log^{d-k+2} n)-time randomized (1+?)-approximation algorithm for this problem with d and ? constant. This is the first algorithm for this problem running in time o(n^k). Its running time is similar to the query time of a data structure recently proposed by Har-Peled, Indyk, and Mahabadi (ICALP\u2718), while not requiring any preprocessing. Up to polylogarithmic factors, it matches a conditional lower bound relying on a conjecture about affine degeneracy testing. In the special case where k = d = O(1), we provide a simple O_?(n^{d-1+?})-time deterministic exact algorithm, for any ? > 0. Finally, we show how to adapt the approximation algorithm for the sparse linear regression and sparse convex regression problems with the same running time, up to polylogarithmic factors