Piecewise Convex Research Articles

Ulam’s Method for Computing Stationary Densities of Invariant Measures for Piecewise Convex Maps with Countably Infinite Number of Branches

Let [Formula: see text] be a piecewise convex map with countably infinite number of branches. In [Góra et al., 2022], the existence of Absolutely Continuous Invariant Measure (ACIM) [Formula: see text] for [Formula: see text] and the exactness of the system [Formula: see text] have been proven. In this paper, we develop an Ulam method for approximation of [Formula: see text], the density of ACIM [Formula: see text]. We construct a sequence [Formula: see text] of maps [Formula: see text] s.t. [Formula: see text] has a finite number of branches and the sequence [Formula: see text] converges to [Formula: see text] almost uniformly. Using supremum norms and Lasota–Yorke-type inequalities, we prove the existence of ACIMs [Formula: see text] for [Formula: see text] with the densities [Formula: see text]. For a fixed [Formula: see text], we apply Ulam’s method with [Formula: see text] subintervals to [Formula: see text] and compute approximations [Formula: see text] of [Formula: see text]. We prove that [Formula: see text] as [Formula: see text] both a.e. and in [Formula: see text]. We provide examples of piecewise convex maps [Formula: see text] with countably infinite number of branches and their approximations by [Formula: see text]’s with finite number of branches. For the increasing values of parameter [Formula: see text] we calculate the errors [Formula: see text].

Nested barycentric coordinate system as an explicit feature map for polyhedra approximation and learning tasks

We introduce a new embedding technique based on a nested barycentric coordinate system. We show that our embedding can be used to transform the problems of polyhedron approximation, piecewise linear classification and convex regression into one of finding a linear classifier or regressor in a higher dimensional (but nevertheless quite sparse) representation. Our embedding maps a piecewise linear function into an everywhere-linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former. We explain the applications of our embedding to the problems of approximating separating polyhedra—in fact, it can approximate any convex body and unions of convex bodies—as well as to classification by separating polyhedra, and to piecewise linear regression.

Invariant measures for random piecewise convex maps

We establish the existence of Lebesgue-equivalent conservative and ergodic σ -finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures around a small neighbourhood of a fixed point where the invariant density functions may diverge. Application covers random intermittent maps with critical points or flat points. We also illustrate that the size of invariant measures tends to infinite for random maps whose right branches exhibit a strongly contracting property on average, so that they have a strong recurrence to a fixed point.

Optimal HVAC Energy and Regulation Reserve Scheduling in Power Markets

This paper considers the optimal scheduling of the hourly energy consumed and regulation reserve capacity offered by a Heating, Ventilation and Air-Conditioning (HVAC) system participating in a day-ahead electricity market. We formulate an Integrated hourly energy and regulation reserve Scheduling and Deployment (ISD) problem using high fidelity models of the HVAC and the building structure's thermal storage property. The ISD problem optimizes the hourly costs and benefits resulting from the power consumed, the regulation reserve capacity offered, the occupant thermal comfort utility, and the expected Intra-Hour Costs (IHCs) due to the HVAC's imperfect tracking of the regulation signal broadcasted by the Independent System Operator every few seconds. Addition of the IHCs to the objective function is the major innovation of this paper. It enables optimal HVAC scheduling to internalize expected regulation signal tracking error cost traded-off against incremental occupant thermal discomfort that may result from perfect tracking. The cost causation circle closes by noting that a high hourly regulation reserve offer may result in higher expected IHCs by increasing the associated tracking error and the incremental occupant discomfort. The paper's innovation is achieved by (i) estimating an analytic function of the hourly HVAC scheduling decisions and state variables that approximates the expected IHCs, and (ii) including this exogenously estimated analytic IHCs function in the objective of the hourly ISD optimization problem. Non-convexities and non-linearities introduced to the HVAC optimization problem resulting from high modeling fidelity and the inclusion of the expected IHCs, are addressed efficiently through piecewise convex relaxations that provide tight optimality bounds. Extensive numerical results are finally provided to demonstrate the applicability and performance of our ISD problem formulation on a realistic office building.

Risk-based distributionally robust optimal dispatch for multiple cascading failures in regional integrated energy system using surrogate modeling

Regional integrated energy system (RIES) with the close coupling of multiple energies faces various cascading failure risks in its operation, and formulating the risk-based optimal dispatch scheme is important to ensure the secure and economic operation of the RIES. Considering that the occurrence probability of each subsequent failure path caused by the initial failure in the cascading failure chain varies with the change of dispatch scheme, the cooling/heating and gas pipeline dynamics and the uncertainty of renewable energy, a risk-based distributionally robust optimal dispatch model for multiple cascading failures in a RIES is proposed. A kernel ambiguity set is constructed by using the maximum mean discrepancy between the real and reference probability distributions and the kernel mean embedding technology, which can obtain the more accurate worst-case probability distribution. Moreover, a surrogate modeling method based on piecewise convex quadratic functions is proposed to learn the mapping between the cascading failure risks and the decision variables, and the original large-scale mixed integer nonlinear programming model is transformed into a small-scale mixed integer quadratic constrained programming model for efficient solution. Finally, case study on an actual RIES demonstrates the effectiveness and advantages of the proposed method. Compared to optimal dispatch without considering the cascading failure risks, the proposed method can reduce the total failure risk cost by nearly 60%.

Intermittent output tracking control of heterogeneous multi-agent systems over wide-area clustered communication networks

In this paper, an output tracking problem is investigated for a wide-area heterogeneous multi-agent system with multiple leaders through intermittent communication, where the communication network consists of clustered subnetworks distributed in a wide area. Firstly, a hierarchical clustered network model is built to describe the communication network topology of heterogeneous agents communicated by an intermittent communication mechanism. Subsequently, dynamic compensators are designed for the heterogeneous agents to estimate the leaders’ state information and a dynamic state feedback hybrid control strategy is designed for each agent to ensure the output tracking consensus. Furthermore, the proposed hybrid control strategy is extended to a case where the complete state information of the agent itself is also not available. Then, under the proposed hybrid control strategy, the closed-loop tracking error system is modeled by a switched system. Consequently, sufficient conditions are derived for the stability of the closed-loop tracking error system under the proposed hybrid control strategy by using a piecewise Lyapunov function and convex synthesis methods. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed hybrid control strategy.

Blind spectral unmixing for characterization of plaque composition based on multispectral photoacoustic imaging

To improve the assessment of carotid plaque vulnerability, a comprehensive characterization of their composition is paramount. Multispectral photoacoustic imaging (MSPAI) can provide plaque composition based on their absorption spectra. However, although various spectral unmixing methods have been developed to characterize different tissue constituents, plaque analysis remains a challenge since its composition is highly complex and diverse. In this study, we employed an adapted piecewise convex multiple-model endmember detection method to identify carotid plaque constituents. Additionally, we explore the selection of the imaging wavelengths in linear models by conditioning the coefficient matrix and its synergy with our unmixing approach. We verified our method using plaque mimicking phantoms and performed ex-vivo MSPAI on carotid endarterectomy samples in a spectral range from 500 to 1300 nm to identify the main spectral features of plaque materials for vulnerability assessment. After imaging, the samples were processed for histological analysis to validate the photoacoustic decomposition. Results show that our approach can perform spectral unmixing and classification of highly heterogeneous biological samples without requiring an extensive fluence correction, enabling the identification of relevant components to assess plaque vulnerability.

Security-Constrained Optimal Traffic-Power Flow With Adaptive Convex Relaxation and Contingency Filtering

The ongoing transportation electrification [e.g., the proliferation of electric vehicles (EVs)] strengthens the interdependence between power and transportation networks and brings tremendous operational challenges to the two critical infrastructure systems. This article proposes an N–1 security-constrained optimal traffic-power flow (SCOTPF) model that includes bilateral contingencies (closure of traffic link and outage of charging station) to coordinate the two networks toward N–1 secure and reliable. The partial user equilibrium (PUE) criterion is adopted to capture the driver’s rerouting behavior in response to the contingency. To facilitate the model tractability, we exploit an adaptive piecewise convex relaxation method based on the convex hull and McCormick envelope to derive a mixed-integer second-order cone programming (MISOCP) model and design a binding contingency identification algorithm to screen redundant contingencies. Finally, the optimal preventive operation strategy is efficiently obtained to reallocate the precontingency traffic-power flow and hence ensures the N–1 reliability and security of the coupled networks. Numerical results on two test systems validate both solution quality and computational efficiency of the SCOTPF. Meanwhile, interdependence between the two networks in contingency scenarios is thoroughly analyzed.

Properties of 3-Triangulations for p-Toroid

In this paper, a method for constructing a toroid and its decomposition into convex pieces is considered. A graph of connection for 3-triangulable toroid is introduced in such a way that these pieces are represented by graph nodes. It is shown that connected, nonorientable graph can serve as a graph of connection for some of the toroids. The relationship between graphs that can be realized on surfaces of different genus and corresponding toroids is considered. Keywords: 3-triangulation of polyhedra, toroids, piecewise convex polyhedra, graph of connection

Spontaneous multi-scale void formation and closure in densifying epithelial and fibroblast monolayers from the sub-confluent state.

Using time-lapse phase contrast microscopy, the formation and closure of spontaneously generated voids in the densifying monolayers of isotropic epithelial cells (ECs) and elongated fibroblast cells (FCs) through proliferation from the sub-confluent state are investigated. It is found that, in both types of monolayers after forming a connected network composed of nematic patches with different orientations, numerous multi-scale voids can be spontaneously formed and gradually close with increasing time. The isotropic fluctuations of deformation and crawling of ECs and the anisotropic axial motion/alignment polarizations of FCs are the two keys leading to the following different generic dynamical behaviors. In EC monolayers, voids exhibit irregular boundary fluctuations and easier cell re-orientation of front layer cells (FLCs) surrounding void boundaries. Void closures are mainly through pinching the gap between the opposite fluctuating void boundaries, and the inward crawling of FLCs to reduce void area associated with topological rearrangement to reduce FLC number. In FC monolayers, large voids have piecewise smooth convex boundaries, and cusp-shaped concave boundaries with cells orienting toward the void at cusp tips. The extension of a thin cell bridge from the cusp tip can bisect a large void into smaller voids. For smaller FC voids dominated by convex boundaries, along which cell alignment prohibits inward crawling, the reduction of FLC number through successive outward squeezing of single FLCs by neighboring FLCs sliding along the void boundary plays an important role for topological rearrangement and void closure. Unlike those surrounding artificial wounds in dense EC monolayers, the absence of ring-like purse-strings surrounding EC and FC voids allows topological rearrangements for reducing void perimeter and void area.

Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

The Cauchy slicings for globally hyperbolic spacetimes and their relation with the causal boundary are surveyed and revisited, starting at the seminal conformal boundary constructions by R. Penrose. Our study covers: (1) adaptive possibilities and techniques for their Cauchy slicings, (2) global hyperbolicity of sliced spacetimes, (3) critical review on the conformal and causal boundaries for a globally hyperbolic spacetime, and (4) procedures to compute the causal boundary of a Cauchy temporal splitting by using isocausal comparison with a static product. New simple counterexamples on $\mathbb{R}^2$ illustrate a variety of possibilities related to these splittings, such as the logical independence (for normalized sliced spacetimes) between the completeness of the slices and global hyperbolicity, the necessity of uniform bounds on the slicings in order to ensure global hyperbolicity, or the insufficience of these bounds for the computation of the causal boundary. A refinement of one of these examples shows that the space of all the (normalized, conformal classes of) globally hyperbolic metrics on a smooth product manifold $\mathbb{R}\times S$ is not convex, even though it is path connected by means of piecewise convex combinations.

No Jackson-type estimates for piecewise q -monotone, q ≥ 3 , trigonometric approximation

UDC 517.5 We say that a function f ∈ C [ a , b ] is q -monotone, q ≥ 3 , if f ∈ C q - 2 ( a , b ) and f ( q - 2 ) is convex in ( a , b ) . Let f be continuous and 2 π -periodic, and change its q -monotonicity finitely many times in [ - π , π ] . We are interested in estimating the degree of approximation of f by trigonometric polynomials which are co- q -monotone with it, namely, trigonometric polynomials that change their q -monotonicity exactly at the points where f does. Such Jackson type estimates are valid for piecewise monotone ( q = 1 ) and piecewise convex ( q = 2 ) approximations. However, we prove, that no such estimates are valid, in general, for co- q -monotone approximation, when q ≥ 3 .

Random Tessellations and Gibbsian Solutions of Hamilton–Jacobi Equations

We pursue two goals in this article. As our first goal, we construct a family $\mathcal{M}_G$ of Gibbs like measures on the set of piecewise linear convex functions $g:\mathbb{R}^2\to\mathbb{R}$. It turns out that there is a one-to-one correspondence between the gradient of such convex functions and $\textit{Laguerre tessellations}$. Each cell in a Laguerre tessellation is a convex polygon that is marked by a vector $\rho\in\mathbb{R}^2$. Each measure $\nu^f\in\mathcal{M}_G$ in our family is uniquely characterized by a kernel $f(x,\rho^-,\rho^+)$, which represents the rate at which a line separating two cells associated with marks $\rho^-$ and $\rho^+$ passes through $x$. To construct our measures, we give a precise recipe for the law of the restriction of our tessellation to a box. This recipe involves a boundary condition, and a dynamical description of our random tessellation inside the box. As we enlarge the box, the consistency of these random tessellations requires that the kernel satisfies a suitable kinetic like PDE. As our second goal, we study the invariance of the set $\mathcal{M}_G$ with respect to the dynamics of such Hamilton-Jacobi PDEs. In particular we $\textit{conjecture}$ the invariance of a suitable subfamily $\widehat{\mathcal{M}_G}$ of $\mathcal{M}_G$. More precisely, we expect that if the initial slope $u_x(\cdot,0)$ is selected according to a measure $\nu^{f}\in \widehat{\mathcal{M}_G}$, then at a later time the law of $u_x(\cdot, t)$ is given by a measure $\nu^{\Theta_t(f)}\in\widehat{\mathcal{M}_G}$, for a suitable kernel $\Theta_t(f)$. As we vary $t$, the kernel $\Theta_t(f)$ must satisfy a suitable kinetic equation. We remark that the function $u$ is also piecewise linear convex function in $(x,t)$, and its law is an example of a Gibbs-like measure on the set of Laguerre tessellations of certain convex subsets of $\mathbb{R}^3$.

Inference for Local Parameters in Convexity Constrained Models

ABSTRACT In this article, we develop automated inference methods for “local” parameters in a collection of convexity constrained models based on the natural constrained tuning-free estimators. A canonical example is given by the univariate convex regression model, in which automated inference is drawn for the function value, the function derivative at a fixed interior point, and the anti-mode of the convex regression function, based on the widely used tuning-free, piecewise linear convex least squares estimator (LSE). The key to our inference proposal in this model is a pivotal joint limit distribution theory for the LS estimates of the local parameters, normalized appropriately by the length of certain data-driven linear piece of the convex LSE. Such a pivotal limiting distribution instantly gives rise to confidence intervals for these local parameters, whose construction requires almost no more effort than computing the convex LSE itself. This inference method in the convex regression model is a special case of a general inference machinery that covers a number of convexity constrained models in which a limit distribution theory is available for model-specific estimators. Concrete models include: (i) log-concave density estimation, (ii) s-concave density estimation, (iii) convex nonincreasing density estimation, (iv) concave bathtub-shaped hazard function estimation, and (v) concave distribution function estimation from corrupted data. The proposed confidence intervals for all these models are proved to have asymptotically exact coverage and oracle length, and require no further information than the estimator itself. We provide extensive simulation evidence that validates our theoretical results. Real data applications and comparisons with competing methods are given to illustrate the usefulness of our inference proposals. Supplementary materials for this article are available online.

Network-Flow-Based Formulations for Convex Hull Pricing With Maximum Start-Ups

Reducing uplift payments has been a challenging problem for most wholesale markets in the U.S. The main difficulty comes from the discrete decision-making and non-linear objective function in the unit commitment (UC) problem. Recently, the convex hull pricing approach has shown promises to reduce the uplift payments, in which efficient algorithms are available when i) the convex hull description for each unit and ii) the convex envelope for the objective function are available. Following this framework, in this paper, we provide network-flow-based compact extended formulations for each UC considering ramping constraints, different initial statuses, and maximum start-up restrictions. Meanwhile, by using a piece-wise linear approximation of a quadratic cost curve in the objective function, we show that the corresponding piece-wise linear convex envelope converges to the quadratic envelope as the number of pieces increases in the convex hull pricing problem. The final computational experiments on a revised IEEE 118-bus system indicates the effectiveness of the proposed approach. The trade-off between the cost-saving and computational time improvement is also reported as a reference for further usage.

Mapping impervious surfaces with a hierarchical spectral mixture analysis incorporating endmember spatial distribution

ABSTRACT Impervious surface mapping is essential for urban environmental studies. Spectral Mixture Analysis (SMA) and its extensions are widely employed in impervious surface estimation from medium-resolution images. For SMA, inappropriate endmember combinations and inadequate endmember classes have been recognized as the primary reasons for estimation errors. Meanwhile, the spectral-only SMA, without considering urban spatial distribution, fails to consider spectral variability in an adequate manner. The lack of endmember class diversity and their spatial variations lead to over/underestimation. To mitigate these issues, this study integrates a hierarchical strategy and spatially varied endmember spectra to map impervious surface abundance, taking Wuhan and Wuzhou as two study areas. Specifically, the piecewise convex multiple-model endmember detection algorithm is applied to automatically hierarchize images into three regions, and distinct endmember combinations are independently developed in each region. Then, spatially varied endmember spectra are synthesized through neighboring spectra using the distance-based weight. Comparative analysis indicates that the proposed method achieves better performance than Hierarchical SMA and Fixed Four-endmembers SMA in terms of MAE, SE, and RMSE. Further analysis suggests that the hierarchical strategy can expand endmember class types and considerably improve the performance for the study areas in general, specifically in less developed areas. Moreover, we find that spatially varied endmember spectra facilitate the reduction of heterogeneous surface material variations and achieve the improved performance in developed areas.

Joint Computation Offloading and Bandwidth Assignment in Cloud-Assisted Edge Computing

Offloading computation based on mobile edge computing paradigms can augment the computational capabilities of resource-scarce mobile devices. However, the capacity limitations of edge servers constrain the performance improvement achieved through computation offloading. In this article, we consider a three-tier computation offloading schema with multiple users, edge servers, and cloud servers. Computation can be offloaded from mobile devices to edge servers or can be further offloaded to remote cloud servers if necessary. Since a number of mobile devices connected to edge servers will share a common wireless communication network, which may contain both uplink and downlink channels, the assignment of bandwidth assignment the channels also constrains the performance improvement of computation offloading. In this paper, the problems of how to determine the offloading strategy and of how to assign the bandwidth are jointly studied and formulated as a programming problem to minimize the average application response time. We analyze the joint problem and further transform it into a piecewise convex programming problem. We propose an efficient algorithm that can find the optimal solution. Extensive experiments demonstrate that our algorithm significantly outperforms previous algorithms. The experimental results also show that the performance of our algorithm is highly robust.

Non-convex isotonic regression via the Myersonian approach

Isotonic regression refers to a class of regression models with order constraints. It is widely used in maximum likelihood estimation of ordered parameter, testing of distributions with ordered means, multistage production systems, and machine learning. A vast majority of the literature considers the isotonic regression problems with convex or piece-wise convex objective functions (or those that can be converted to such functions). We connect a class of isotonic regression problems with the so-called ironing problem in mechanism design, by establishing a discrete version of the Myerson’s ironing method. We use such a connection to solve an isotonic regression problem with non-convex objective functions. We also prove the optimality of pool adjacent violator (PAV) algorithm in such a case.

Cost-Efficient Resource Provisioning for Dynamic Requests in Cloud Assisted Mobile Edge Computing

Mobile edge computing is emerging as a new computing paradigm that provides enhanced experience to mobile users via low latency connections and augmented computation capacity. As the amount of user requests is time-varying, while the computation capacity of edge hosts is limited, Cloud Assisted Mobile Edge (CAME) computing framework is introduced to improve the scalability of the edge platform. By outsourcing mobile requests to clouds with various types of instances, the CAME framework can accommodate dynamic mobile requests with diverse quality of service requirements. In order to provide guaranteed services at minimal system cost, the edge resource provisioning and cloud outsourcing of the CAME framework should be carefully designed in a cost-efficient manner. Specifically, two fundamental issues should be answered: (1) what is the optimal edge computation capacity configuration? and (2) what types of cloud instances should be tenanted and what is the amount of each type? To solve these issues, we formulate the resource provisioning in CAME framework as an optimization problem. By exploiting the piecewise convex property of this problem, the Optimal Resource Provisioning (ORP) algorithms with different instances are proposed, so as to optimize the computation capacity of edge hosts and meanwhile dynamically adjust the cloud tenancy strategy. The proposed algorithms are proved to be with polynomial computational complexity. To evaluate the performance of the ORP algorithms, extensive simulations and experiments are conducted based on both the widely-used traffic models and the Google cluster usage tracelogs, respectively. It is shown that the proposed ORP algorithms outperform the local-first and cloud-first benchmark algorithms in system flexibility and cost-efficiency.

Decay of correlations and memory loss for Lasota–Yorke convex maps

For a class of piecewise convex maps f on the interval , we show that f has a unique absolutely continuous invariant probability measure μ with exponential decay of correlations, and we also present the explicit upper bounds on the rate. Moreover, we show the exponential loss of memory for a sequential dynamical system consisting of piecewise convex maps.