Local Convexity Research Articles

Solving a class of nonsmooth resource allocation problems with directed graphs through distributed Lipschitz continuous multi-proximal algorithms

In this paper, two distributed multi-proximal primal–dual algorithms are proposed to deal with a class of distributed nonsmooth resource allocation problems. In these problems, the global cost function is the summation of local convex and nonsmooth cost functions, each of which consists of one twice differentiable function and multiple nonsmooth functions. Communication graphs of underlying multi-agent systems are directed and strongly connected but not necessarily weighted-balanced. The multi-proximal splitting is introduced to deal with the difficulty caused by the unproximable property of the summation of those nonsmooth functions. Moreover, it can also guarantee the Lipschitz continuity of proposed algorithms. Auxiliary variables in the multi-proximal splitting are designed to estimate subgradients of nonsmooth functions. Theoretically, the convergence analysis is conducted by employing Lyapunov stability theory and integral input-to-state stability (iISS) theory with respect to sets. It shows that proposed algorithms converge to the optimal point while satisfying resource allocation constraints.

First Satellite Telemetry Study of Movement Behavior of Juvenile Golden Eagles from Mexico

ABSTRACT The southern limit of the Golden Eagle's (Aquila chrysaetos) breeding range in North America is Mexico, where the eagle is the national symbol yet designated as a threatened, high priority species for conservation action. Movement information needed for conserving Mexico's Golden Eagles is sparse; knowledge of dispersal from natal areas is essential to understand the eagle's ecology and help provide for its management. Using satellite telemetry data, we analyzed movements of three males and one female from central Mexico during their first year of life; we documented (1) timing and distance of initial dispersal movements, (2) total distance traveled and maximum distance from natal site by month of age following fledging, and (3) size of areas (based on 95% adaptive local convex hulls) across which eagles ranged following initial dispersal. Individual eagles dispersed from their natal areas between mid-September and mid-November, at 6–8 mo of age. Monthly total distance traveled by males reached approximately 350–1350 km at 8–11 mo; the female's peak monthly travel was 3000 km, at age 7 mo. Monthly proximity to natal sites by individuals at ages 8–12 mo was relatively constant, averaging 17.9 km (SD = 5.7) to 129.1 km (SD = 11.3). After dispersal, the monthly ranging areas overall increased during the first year of life for all eagles, especially the female, due mainly to multiple long-distance excursions. Our data suggest that movement behavior of juvenile Golden Eagles from Mexico is mostly similar to that of conspecifics from nonmigratory populations elsewhere. Our study may help serve as a foundation for future work to better understand movement dynamics and resource selection by Mexico's Golden Eagles.

Deformation Characteristics in a Stretch-Based Dimensional Correction Method for Open, Thin-Walled Extrusions

Dimensional accuracy of incoming components is crucial for automated welding and assembly in mass volume production. However, thin-walled extrusions made to industrial standards show severe dimensional variations, including gap opening, sidewall inclination, local convexity, and so on. Thus, one major challenge is to provide a low-cost correction method to improve the dimensional accuracy at a level demanded by automated assembly and/or product fit-up. A novel correction method called transverse stretch and local bending (TSLB) has recently been developed, enabling one to efficiently correct the dimensional deviations in thin-walled, U-channel profiles at a low cost. However, the lack of in-depth understanding of the underlying mechanism makes it challenging to efficiently optimise and control the process. In this study, the feasibility of this new technique was experimentally validated by four groups of TSLB tests with different profile dimensions, showing a dimensional accuracy improvement of about 92% compared with the as-received parts. The evolution of the critical dimensional characteristics, including gap opening and bottom convexity, is analysed numerically throughout four stages consisting of inserting, releasing, calibration, and springback. It is found that the inserting stage greatly reduces the dimensional deviations in a pure bending state, while the calibration stages further minimise the deviations in the bending and transverse stretching combined state. In addition, the wedge angle of the tool is found to be critical to the dimensional accuracy improvement. The low wedge angle facilitates the correction of sidewall inclination and gap opening, while the high wedge angle contributes to mitigating bottom convexity. The overall outcome of this study enhances the fundamental understanding of the effects of in-process stretching and local-bending on the dimensional capabilities of U-channel extrusions. This can ultimately generate guidelines that will lead to new application areas of aluminium extrusions in highly competitive marketplaces.

A Privacy-Masking Learning Algorithm for Online Distributed Optimization over Time-Varying Unbalanced Digraphs

This paper investigates a constrained distributed optimization problem enabled by differential privacy where the underlying network is time-changing with unbalanced digraphs. To solve such a problem, we first propose a differentially private online distributed algorithm by injecting adaptively adjustable Laplace noises. The proposed algorithm can not only protect the privacy of participants without compromising a trusted third party, but also be implemented on more general time-varying unbalanced digraphs. Under mild conditions, we then show that the proposed algorithm can achieve a sublinear expected bound of regret for general local convex objective functions. The result shows that there is a trade-off between the optimization accuracy and privacy level. Finally, numerical simulations are conducted to validate the efficiency of the proposed algorithm.

Seasonal variation in home range size of evening bats (Nycticeius humeralis) in an urban environment

Abstract Despite the negative impacts of urbanization on bats, green spaces in urban environments, such as parks, cemeteries, and golf courses, have the potential to provide resources necessary for these animals. Water resources in these areas include natural or semi-natural ponds, streams, and drainage ditches. Such water resources, however, are frequently ephemeral when subject to prolonged periods of high temperatures and low precipitation. We hypothesized that home ranges of evening bats (Nycticeius humeralis) would increase in size or shift from urban green spaces into surrounding neighborhoods to access alternative resources, such as residential swimming pools, when water resources in green spaces are scarce. To explore whether seasonal variations in bat home ranges occurred, we radio-tracked resident evening bats in a local park system during their summer activity period 2017–2019 in Fort Worth, Texas, USA. We used Local Convex Hulls created by associating point locations with their nearest neighbor to estimate home range size, location, and the percentage area that fell within the surrounding neighborhood. We compared these variables to temperature and precipitation using various regression models. We successfully tracked 30 bats over the 3-year period and found a positive correlation between home range size and temperature. Home range sizes increased 6-fold when temperatures exceeded 30°C. Our study indicates the importance of urban neighborhoods surrounding green spaces in providing alternative resources, such as water, for bats. If managed appropriately, these urban areas have the potential to act as oases for bat populations.

Analysis of the deformation mechanism and stability of bedding soft rock slope

The lithology of the inlet side slope of the diversion tunnel on the right bank of Wudongde Hydropower Station is dominated by thin marbled dolomite. During construction, numerous cracks form along the elevation direction, and large monitoring deformation values appear in the consequent slope section. In this study, engineering geological analysis, deformation monitoring data analysis, and three-dimensional numerical simulation are combined to investigate the formation mechanism of cracks, the range of local deformed bodies, and the deformation and failure mechanism of deformed bodies. The stability of the slope is rechecked via the rigid-body limit equilibrium method based on the above analysis. Results show that the local convex terrain in the consequent slope section is cut through unfavorable structural surfaces and form a deformable body. Under the influence of unfavorable factors, such as the bedding cut foot caused by the lower excavation unloading and rainfall, the deformation of the shallow slope surface continuously increases; as a result, the shotcrete layer cracks. The deformation force comes from the middle and rear parts of the slope, and the instability mode slides along the bottom boundary level. After prestressed cables are added, the safety factor of the deformed part can satisfy the code requirements, and the existence of the deformed part does not affect the overall slope stability. Therefore, this study can provide technical support for engineering construction and a reference for similar slope engineering research.

Distributed Bandit Online Convex Optimization With Time-Varying Coupled Inequality Constraints

Distributed bandit online convex optimization with time-varying coupled inequality constraints is considered, motivated by a repeated game between a group of learners and an adversary. The learners attempt to minimize a sequence of global loss functions and at the same time satisfy a sequence of coupled constraint functions, where the constraints are coupled across the distributed learners at each round. The global loss and the coupled constraint functions are the sum of local convex loss and constraint functions, respectively, which are adaptively generated by the adversary. The local loss and constraint functions are revealed in a bandit manner, i.e., only the values of loss and constraint functions are revealed to the learners at the sampling instance, and the revealed function values are held privately by each learner. Both one- and two-point bandit feedback are studied with the two corresponding distributed bandit online algorithms used by the learners. We show that sublinear expected regret and constraint violation are achieved by these two algorithms, if the accumulated variation of the comparator sequence also grows sublinearly. In particular, we show that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(T^{\theta })$</tex-math></inline-formula> expected static regret and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(T^{7/4-\theta })$</tex-math></inline-formula> constraint violation are achieved in the one-point bandit feedback setting, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(T^{\max \lbrace \kappa,1-\kappa \rbrace })$</tex-math></inline-formula> expected static regret and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(T^{1-\kappa /2})$</tex-math></inline-formula> constraint violation in the two-point bandit feedback setting, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta \in (3/4,5/6]$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\kappa \in (0,1)$</tex-math></inline-formula> are user-defined tradeoff parameters. Finally, the tightness of the theoretical results is illustrated by numerical simulations of a simple power grid example, which also compares the proposed algorithms to algorithms existing in the literature.

Pimples reduce and dimples enhance flat dielectric surface image repulsion.

In solid-liquid, or liquid-liquid, interfaces with dielectric contrast, charged particles interact with the induced polarization charge of the interface. These interactions contribute to an effective self-energy of the bulk ions and mediate ion-ion interactions. For flat interfaces, the self-energy and the mediated interactions are neatly constructed by the image charge method. For other geometries, explicit results are scarce and the problem must be treated via approximations or direct computation. The case of interfaces with roughness is of great practical importance. This article provides analytical results, valid to first-order in perturbation theory, for the self-energy of particles near rough substrates. Explicit formulas are provided for the case of a sinusoidal deformation of a flat surface. Generic deformations can be treated by superposition. In addition to results for the self-energy, the surface polarization charge is presented as a quadrature. The interaction between an ion and the deformed surface is modified by the change in relative distance as well as by the local curvature of the surface. Solid walls, with a lower dielectric constant than the liquid, repel all ions. We show that the repulsion is reduced by local convexity and enhanced by concavity; dimples are more repulsive than pimples.

Quantitative comparisons of multiscale geometric properties

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface cube $R$, \[ \sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d} \lesssim ({\rm diam} R)^{d}.\] We show that, for lower content regular sets that aren't necessarily Ahlfors regular, if $\beta_{E}(R)$ denotes the square sum of $\beta$-numbers over subcubes of $R$ as in the Traveling Salesman Theorem for higher dimensional sets [AS18], then \[ \mathscr{H}^{d}(R)+\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d}\sim \beta_{E}(R). \] We prove similar results for other uniform rectifiability critera, such as the Local Symmetry, Local Convexity, and Generalized Weak Exterior Convexity conditions. En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes.

Random Sleep Scheme-Based Distributed Optimization Algorithm Over Unbalanced Time-Varying Networks

This article considers a category of constrained convex optimization problems over multiagent networks. The networked agents aim at collaboratively minimizing the sum of all locally known objective functions over a common convex set. Each agent possesses only its local convex function and its state is constrained to a privately known convex set. A novel distributed algorithm is proposed over time-varying unbalanced directed networks based on epigraph form of the original optimization problem and consensus theory. By incorporating the random sleep scheme, the proposed algorithm allows each agent to independently and randomly decide whether to calculate subgradient and take projection at each iteration, which alleviates the cost of subgradient observation. Besides, it neither resorts to doubly stochastic weight matrices (but only row-stochastic) nor the information of the graph sequence to execute. The convergence of the algorithm is explicitly analyzed under conditions that the sequence of time-varying directed graphs is uniformly jointly strongly connected and the subgradients of all local objective functions are bounded over a convex set. The optimization algorithm ensures zero-gap on the expected distance between the estimated value of each agent and the exact optimal solution. The two simulation cases are presented to demonstrate the practicability of the algorithm and correctness of the obtained theoretical results.

Convergence of Distributed Accelerated Algorithm Over Unbalanced Directed Networks

In this article, the problem of the distributed convex optimization is investigated, where the target is to collectively minimize a sum of local convex functions over an unbalanced directed multiagent network. Each agent in the network possesses only its private local objective function, and the sum of all local objective functions constitutes the global objective function. We particularly consider the scenario, where the underlying interaction network is strongly connected and the relevant weight matrix is row stochastic. To collectively figure out the optimization problem, a distributed accelerated convergence algorithm where agents utilize uncoordinated step-sizes is presented by incorporating consensus of multiagent networks into distributed inexact gradient tracking technique. Most of the existing methods require all agents to possess the out-degree information of their in-neighbors, which is impractical and hardly inevitable as interpreted in this article. By utilizing the small-gain theorem, we prove that if the maximum step-size is positive and sufficiently small (constrained by a specific upper bound), the proposed algorithm, termed as SGT-FROST, converges geometrically to the optimal solution given that the objective functions are smooth and strongly convex. A certain convergence rate is also shown. Simulations confirm the findings in this article.

Model-Assisted Uniformly Honest Inference for Optimal Treatment Regimes in High Dimension

This article develops new tools to quantify uncertainty in optimal decision making and to gain insight into which variables one should collect information about given the potential cost of measuring a large number of variables. We investigate simultaneous inference to determine if a group of variables is relevant for estimating an optimal decision rule in a high-dimensional semiparametric framework. The unknown link function permits flexible modeling of the interactions between the treatment and the covariates, but leads to nonconvex estimation in high dimension and imposes significant challenges for inference. We first establish that a local restricted strong convexity condition holds with high probability and that any feasible local sparse solution of the estimation problem can achieve the near-oracle estimation error bound. We further rigorously verify that a wild bootstrap procedure based on a debiased version of the local solution can provide asymptotically honest uniform inference for the effect of a group of variables on optimal decision making. The advantage of honest inference is that it does not require the initial estimator to achieve perfect model selection and does not require the zero and nonzero effects to be well-separated. We also propose an efficient algorithm for estimation. Our simulations suggest satisfactory performance. An example from a diabetes study illustrates the real application. Supplementary materials for this article are available online.

Distinguished Property in Tensor Products and Weak* Dual Spaces

A local convex space E is said to be distinguished if its strong dual Eβ′ has the topology β(E′,(Eβ′)′), i.e., if Eβ′ is barrelled. The distinguished property of the local convex space CpX of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space CpX is distinguished if and only if any function f∈RX belongs to the pointwise closure of a pointwise bounded set in CX. The extensively studied distinguished properties in the injective tensor products CpX⊗εE and in Cp(X,E) contrasts with the few distinguished properties of injective tensor products related to the dual space LpX of CpX endowed with the weak* topology, as well as to the weak* dual of Cp(X,E). To partially fill this gap, some distinguished properties in the injective tensor product space LpX⊗εE are presented and a characterization of the distinguished property of the weak* dual of Cp(X,E) for wide classes of spaces X and E is provided.

RISK FACTORS FOR MALIGNANCY OF PAPILLOMAS IN THE OROPHARYNGEAL MUCOUS MEMBRANE ASSOCIATED WITH HUMAN PAPILLOMAVIRUS (HPV)

Analysis of the HPV positive papilloma structures at different levels of pathological process development was carried out. Classical staining of preparations with hematoxylin and eosin was used for immunohistochemical determination of Ki67-positive cells and phenotyping of CD positive cells. We found that the process of papilloma formation begins with a local increase in the proliferative activity of keratinocytes which contributes to the formation of a local epithelial convex above the surface of the oropharyngeal mucosa (OPM) in the form of papilloma. The connective tissue of the OPM adjacent to the epithelium grows at the second stage. At the same time here is a decrease in immunocytes/macrophages number of the oral epithelium not only in the growth zone of the papilloma but in the adjacent neoplasm tissue. The third stage is characterized by the destruction of the basal membrane of the OPM. Apoptotic cells in the cambial layer and forming leukocytes infiltrate the OPM lamina propria. Virus-infected keratinocytes are phagocytized by macrophages or exfoliated from the surface of the epithelial layer. The emerging in the middle layers defect of tissue of the epithelial layer and the absence of Langerhans cells indicate a relationship between the migration of antigen-presenting cells expressing CD68 with impaired differentiation and specialization of keratinocytes. The conclusion is based on the analogy of "leukemic failure" in leukemia and on the absence of differentiating epithelial cells between the cambium and the specialized surface layer in the long-existing papilloma. The disappearance from the epithelium of CD68-positive cells specific to the epithelial layer is a prognostic sign of malignization in the mucous membrane of the oropharynx.

Reconstruction of Complex Roof Semantic Structures from 3D Point Clouds Using Local Convexity and Consistency

Three-dimensional (3D) building models are closely related to human activities in urban environments. Due to the variations in building styles and complexity in roof structures, automatically reconstructing 3D buildings with semantics and topology information still faces big challenges. In this paper, we present an automated modeling approach that can semantically decompose and reconstruct the complex building light detection and ranging (LiDAR) point clouds into simple parametric structures, and each generated structure is an unambiguous roof semantic unit without overlapping planar primitive. The proposed method starts by extracting roof planes using a multi-label energy minimization solution, followed by constructing a roof connection graph associated with proximity, similarity, and consistency attributes. Furthermore, a progressive decomposition and reconstruction algorithm is introduced to generate explicit semantic subparts and hierarchical representation of an isolated building. The proposed approach is performed on two various datasets and compared with the state-of-the-art reconstruction techniques. The experimental modeling results, including the assessment using the International Society for Photogrammetry and Remote Sensing (ISPRS) benchmark LiDAR datasets, demonstrate that the proposed modeling method can efficiently decompose complex building models into interpretable semantic structures.

On local hulls of Levi-flat hypersurfaces

We discuss local polynomial convexity of real analytic Levi-flat hypersurfaces in [Formula: see text], [Formula: see text], near singular points.

An outer-approximation guided optimization approach for constrained neural network inverse problems

This paper discusses an outer-approximation guided optimization method for constrained neural network inverse problems with rectified linear units. The constrained neural network inverse problems refer to an optimization problem to find the best set of input values of a given trained neural network in order to produce a predefined desired output in presence of constraints on input values. This paper analyzes the characteristics of optimal solutions of neural network inverse problems with rectified activation units and proposes an outer-approximation algorithm by exploiting their characteristics. The proposed outer-approximation guided optimization comprises primal and dual phases. The primal phase incorporates neighbor curvatures with neighbor outer-approximations to expedite the process. The dual phase identifies and utilizes the structure of local convex regions to improve the convergence to a local optimal solution. At last, computation experiments demonstrate the superiority of the proposed algorithm compared to a projected gradient method.

Cluster-based distributed augmented Lagrangian algorithm for a class of constrained convex optimization problems

We propose a distributed solution for a constrained convex optimization problem over a network of clustered agents each consisted of a set of subagents. The communication range of the clustered agents is such that they can form a connected undirected graph topology. The total cost in this optimization problem is the sum of the local convex costs of the subagents of each cluster. We seek a minimizer of this cost subject to a set of affine equality constraints, and a set of affine inequality constraints specifying the bounds on the decision variables if such bounds exist. We design our distributed algorithm in a cluster-based framework which results in a significant reduction in communication and computation costs. Our proposed distributed solution is a novel continuous-time algorithm that is linked to the augmented Lagrangian approach. It converges asymptotically when the local cost functions are convex and exponentially when they are strongly convex and have Lipschitz gradients. Moreover, we use an ϵ-exact penalty function to address the inequality constraints and derive an explicit lower bound on the penalty function weight to guarantee convergence to ϵ-neighborhood of the global minimum value of the cost. A numerical example demonstrates our results.

Navigable Space Construction from Sparse Noisy Point Clouds

Robot navigation with noisy perception is a fundamental but challenging task. We present a framework for creating navigable space from cluttered point clouds generated by low-end sensors with high sparsity and noise. Our method incrementally seeds and creates local convex regions free of obstacle points along robot's trajectory. Then a dense version of the point cloud is reconstructed through a map point regulation process where the original noisy map points are first projected onto a series of local convex hull surfaces, after which those points falling inside the convex hulls are culled. We have tested our proposed framework using a public autonomous driving dataset and a manually built structured environment for various performance evaluation, as well as inside our research building for ground robot navigation tasks. Our results reveal that the reconstructed navigable space has small volume loss (error) comparing with the ground truth, and the method is highly efficient, allowing real-time navigation computation.

Translating solutions to the Gauss curvature flow with flat sides

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.