Original Problem Research Articles

Relaxation approach for optimization of free boundary problems

AbstractWe consider an optimal control problem (OCP) constrained by a free boundary problem (FBP). FBPs have various applications such as in fluid dynamics, flow in porous media or finance. For this work we study a model FBP given by a Poisson equation in the bulk and a Young‐Laplace equation accounting for surface tension on the free boundary. Transforming this coupled system to a reference domain allows to avoid dealing with shape derivatives. However, this results in highly nonlinear partial differential equation (PDE) coefficients, which makes the OCP rather difficult to handle. Therefore, we present a new relaxation approach by introducing the free boundary as a new control variable, which transforms the original problem into a sequence of simpler optimization problems without free boundary. In this paper, we formally derive the adjoint systems and show numerically that a solution of the original problem can be indeed asymptotically approximated in this way.

Relaxation approaches for nonlinear sparse optimization problems

In many recent applications, sparse solutions of the optimization problems are favoured over non-sparse solutions with comparable objective values. A standard method to induce the sparsity of the solution is based on the use of the norm in the objective. However, if the underlying optimization problem is nonlinear, the solution of the nonlinear (sparse) -optimization problem is difficult. Therefore, it is often approximated using the convex -norm although this can lead to suboptimal solutions for the sparsity of the solution. In this paper, we follow another direction. We present exact reformulations (with respect to the norm) and their relaxations leading to standard nonlinear but nonconvex programmes. We discuss and relate the relations between the different reformulations with repect to the original problem. We accompany our theoretical results by some numerical tests using randomly generated datasets.

A partition-based problem transformation algorithm for classifying imbalanced multi-label data

Multi-label learning has garnered much research interest due to its wide range of real-world applications. Many multi-label learning methods have been proposed; however, few have addressed the class imbalance problem existing in multi-label data. Even though some studies have taken this issue into account, most of them have ignored the label correlations or only considered random correlations between them. In this study, we propose a novel partition-based imbalanced multi-label learning algorithm, named Multi-label Learning based on Hierarchical Clustering (MLHC), to tackle this problem. MLHC first carries out hierarchical clustering on the original label space to divide it into several disconnected subspaces, each of which contains several labels that are strongly correlated with each other. Then, for each label subspace, we use the problem transformation strategy to convert it into a multi-class problem by binary coding. Any multi-class imbalance learning algorithm can be applied to the transformed multi-class data. Finally, the classification results will be decoded to retrieve the corresponding label subspace, and all label subspace results are combined to show the predicted label vector in the original label space. We conducted experiments not only on thirteen benchmark multi-label datasets but also carried out them on XJTU-SY which is a multi-label engineering application dataset, and the results indicated that our proposed MLHC learning algorithm outperforms several state-of-the-art class imbalance multi-label learning algorithms, demonstrating the effectiveness and necessity of discovering label correlations and transforming the original imbalanced multi-label learning problem into multiple strongly correlated multi-class imbalanced learning problems.

Observer-based event-triggered recursive optimal tracking control for a class of strict-feedback nonlinear systems

In this paper, an innovative event-triggered optimal tracking control algorithm is proposed for input saturated strict-feedback nonlinear systems with unknown dynamics. In order to reduce the requirement of configuring a complete suit of sensors and enhance the reliability of the controlled system, a neural networks (NNs) based adaptive state observer is developed firstly to reconstruct the system states. Subsequently, based on the state estimation information, a hybrid-triggered feedforward controller is designed to transform the original tracking control problem into an equivalent regulation issue, which is then solved by developing an event-triggered optimal controller. Therefore, the final controller consists of a hybrid-triggered feedforward controller and an event-triggered optimal controller. In order to make the actual input signals of the two controllers be updated simultaneously, a synchronization-oriented triggering rule is established by using multiple triggering errors. By virtue of this unique framework, the proposed control scheme can not only minimize the predefined cost function, but also greatly reduce the data transmission. What is more, the convergence properties of the proposed control strategy are achieved by using Lyapunov theory. It is important to note that unlike the widely adopted observer-controller framework, where the separation principle holds for the design of the state observer, there is a considerable coupling relationship between the error dynamics of the state observer and the event-triggered optimal controller in this paper. The distinguishing feature of the proposed method is its ability to ensure a satisfactory level of precision in both state estimation and tracking control, even in the presence of control saturation issues. At last, the proposed control strategy is applied to the tracking control problem of a high-order robot system and marine surface vehicle to demonstrate its effectiveness.

Efficient iterative optimization to real-time train regulation in urban rail transit networks combined with Benders decomposition method

Considering the rapid expansion of urban rail transit networks and increasing transfer demands, this paper investigates the design of real-time train regulation strategies for urban rail networks under disturbances. Specifically, a mixed-integer nonlinear programming (MINLP) model is constructed in a rolling horizon (RH) framework to reduce train deviations and passenger waiting times, which involves complex coupling relationships among network-wide non-transfer passengers, transfer passengers and train traffics of different lines. To effectively address the model complexity and adapt to the real-time nature of this issue, we proposed an efficient iterative optimization (IO) approach to split the original problem into a smaller-scale regulation issue and a computationally cheap passenger loading procedure, which can efficiently treat the introduced nonconvexity of the exact modelling scheme. Besides, tailored to the mixed-integer property of the model, the generalized Benders decomposition (GBD) technique is incorporated into our algorithm. Furthermore, to verify the effectiveness of our method, we describe the detailed implementation of a series of numerical experiments on a small-scale and a real-world case. Computational results show that our regulation method can effectively contribute to the reliability of train systems, improving operational efficiency and service level. Besides, it performs promising computational efficiency in dealing with large-scale real-world cases, applicable for real-time applications.

On the numerical integration of singular initial and boundary value problems for generalised Lane–Emden and Thomas–Fermi equations

We propose a geometric approach for the numerical integration of singular initial and boundary value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane–Emden equations and the Thomas–Fermi equation.

A Minimum Joint Error Entropy-Based Localization Method in Mixed LOS/NLOS Environments

In this paper, we address the time-of-arrival (TOA) based source localization problem in mixed light-of-sight (LOS)/non-light-of-sight (NLOS) environments, where localization accuracy is degraded by both measurement noise and NLOS error. A novel exponential optimization problem is formulated based on new minimum joint error entropy criteria and statistical characteristics of measurement noise. After that, a two-step relaxation method is proposed. In the first step, the original problem is relaxed into a non-exponential problem which maintains the consistency of the solution. The second step is to transform the non-exponential problem into a convex problem. Furthermore, we extend the method to asynchronous networks where the source and anchors are not time synchronized. Numerical results show that the proposed method can provide significant robust performance in synchronous or asynchronous networks, whether in sparse or dense NLOS scenarios.

Age-of-Information-Based Computation Offloading and Transmission Scheduling in Mobile-Edge-Computing-Enabled IoT Networks

The emergence of Mobile Edge Computing (MEC) technology has deployed edge clouds with strong computing capabilities closer to Internet of Thing (IoT) devices, which can effectively meet the demands for computing power and latency. However, in addition to the stringent latency requirements, more and more emerging IoT applications also have higher standards for the freshness and timeliness of collected information. In order to ensure the freshness and high information value in IoT system, we propose an Age of Information (AoI)-based optimization strategy for computation offloading and transmission scheduling. The strategy considers the AoI during the transmission phase and the execution phase, respectively, under the constraints of delay and remaining energy. Then, a joint optimization model is established based on the comprehensive benefits of AoI and computation rate. To address the strong coupling between the offloading decision and the transmission decision, the original optimization problem is divided into two stages. By the use of the Deep Deterministic Policy Gradient (DDPG) algorithm and the Dueling Double Deep Q Network (D3QN) algorithm, the solution is obtained in terms of the offloading decision and transmission scheduling decision, respectively. The proposed joint optimization strategy considers the impact of the transmission decision on the offloading decision and is adaptable to the dynamic changes in the channel connection between the edge cloud and the user due to user mobility. Experimental results show that compared with other offloading and transmission strategies, the proposed approach has higher overall system revenue and lower AoI.

An Energy-Efficient Multimode Transmission Scheme for Underwater Sensor Network

Underwater sensor network plays a critical role in the ocean data collection owing to its capability of providing reliable and wide communication coverage. How to assure lifetime performance of the network with limited energy supply has always been a key research issue. In this paper, we propose an underwater acoustic sensor network model which supports data delivery from each underwater sensor node (USN) to sea surface sink node (SN) in either direct or relay-assisted transmission mode, i.e., transmission mode between each USN and the SN can be dynamically selected according to network’s current state. To optimize its lifetime performance, we model the mode selection and resource allocation issue jointly into a non-convex and mixed integer programming problem. In order to efficiently solve it, we further divide the original problem into two subproblems: resource allocation subproblem and joint mode and relay selection subproblem. We prove that the reformulated non-convex resource allocation problem can be equivalent to a convex optimization problem, and its optimal solution can be found by using Lagrange dual decomposition method. For the second subproblem, some critical conditions are analyzed to determine optimal relays with the criterion of balancing energy consumption between USNs and a matching approach is designed to obtain the mode and relay selection results based on USNs’ priorities. Simulation results validate that the proposed transmission scheme is effective and superior to the traditional time division multiple access scheme.

Efficient Method for Wet Modal Analysis of Cyclic Periodic Fluid/Structure Systems

In this paper, we propose an efficient method for the modal analysis of vertical cyclic periodic tanks partially filled with liquid. The fluid effect on the vibration of the structure is modeled by introducing an added mass matrix. Because the added mass matrix formed by the boundary element method is fully populated, it requires considerable computer memory when the wet modal analysis involves substantial wetted elements. Additionally, the dense added mass matrix will destroy the sparsity of the total mass matrix, resulting in a significant computational burden for solving the eigenvalue problem. Based on the cyclic periodic property of the fluid/structure system, it is proved that the added mass matrix of the vertical cyclic periodic tank also has a block-circulant property. Therefore, according to the group representation theory, the original eigenvalue problem of the wet modal analysis can be decoupled into a series of independent subproblems. Because the computational scale of the subproblems is much smaller than that of the original eigenvalue problem, the proposed method can significantly reduce the computational cost. Numerical examples show that the proposed method can solve the modal analysis of the cyclic periodic fluid/structure system with high accuracy and efficiency.

The Airy equation with nonlocal conditions

AbstractWe study a third‐order dispersive linear evolution equation on the finite interval subject to an initial condition and inhomogeneous boundary conditions but, in place of one of the three boundary conditions that would typically be imposed, we use a nonlocal condition, which specifies a weighted integral of the solution over the spatial interval. Via adaptations of the Fokas transform method (or unified transform method), we obtain a solution representation for this problem. We also study the time periodic analog of this problem, and thereby obtain long time asymptotics for the original problem with time periodic boundary and nonlocal data.

Unimodular Waveform Design for the DFRC System with Constrained Communication QoS

In this study, we investigated two waveform design problems for a dual-functional radar communication (DFRC) system, taking into consideration different constrained communication quality-of-service (QoS) requirements. Our objective was to minimize the mean-square error (MSE) of radar beampattern matching as the cost function. To this end, the multi-user interference (MUI) energy constraint and constructive interference (CI) constraint were, respectively, formulated to ensure the communication QoS. It is important to note that we designed a strict per-user MUI energy constraint at each sampling moment to achieve more accurate control over communication performance. Additionally, we introduced a constant-modulus constraint to optimize the efficiency of the radio frequency (RF) amplifier. To tackle the nonconvex waveform design problems encountered, we employed the alternative direction methods of multipliers (ADMM) technique. This allowed us to decompose the original problem into two solvable subproblems, which were then solved using the majorization–minimization (MM) method and geometrical structure. Finally, we obtained extensive simulation results which demonstrate the effectiveness and superiority of the proposed algorithm.

A Central Scheme for Two Coupled Hyperbolic Systems

AbstractA novel numerical scheme to solve two coupled systems of conservation laws is introduced. The scheme is derived based on a relaxation approach and does not require information on the Lax curves of the coupled systems, which simplifies the computation of suitable coupling data. The coupling condition for the underlying relaxation system plays a crucial role as it determines the behaviour of the scheme in the zero relaxation limit. The role of this condition is discussed, a consistency concept with respect to the original problem is introduced, the well-posedness is analyzed and explicit, nodal Riemann solvers are provided. Based on a case study considering the p-system of gas dynamics, a strategy for the design of the relaxation coupling condition within the new scheme is provided.

Optimization of train schedule with uncertain maintenance plans in high-speed railways: A stochastic programming approach

In high-speed railways, unexpected disturbances on maintenance activities may cause serious delays of the scheduled trains and greatly affect the service quality for traveling passengers. In contrast to most existing studies that focused on deterministic maintenance activities, this paper develops a two-stage stochastic programming approach to address the optimization of train schedules under uncertain maintenance plans. Specifically, in the first stage, we aim to determine the departure times of trains from the origin station, since this information needs to be public to passengers in advanced. The objective function is to minimize the expected travel time of trains under uncertain duration time of maintenance activities. In the second stage, given the specific information of maintenance activities, we generate the train schedule by adjusting the stop patterns, train orders and the assignment of tracks at key stations. Due to the computational difficulties arising from the large number of discrete decision variables, we particularly develop a dual decomposition based solution approach to solve the two-stage stochastic model. Our approach decomposes the original problem into a set of scenario-dependent subproblems with much fewer number of variables, which greatly improves the computational efficiency. Finally, we conduct several sets of real-world instances based on the Beijing–Guangzhou high-speed railway corridor to verify the effectiveness of the proposed model and solution approach. The results demonstrate that our approach evidently outperforms state-of-art solvers (Gurobi), especially for large-scale instances that Gurobi cannot even return feasible solutions.

A survivable variant of the ring star problem

AbstractThe Ring Star Problem consists in selecting a subset of nodes called hubs including the depot and linking them with a cycle, the remaining nodes being connected to exactly one hub, at minimum cost. We study a survivable variant of the Ring Star Problem where at most one node in a given subset of so‐called uncertain nodes can fail if selected as a hub. We model this problem as an Integer Linear Program (ILP), that is also addressed with a Branch‐and‐Benders‐cut decomposition. The Benders subproblem is turned into a linear program with the addition of new inequalities that are shown to be facet‐defining, and several enhancements to both the ILP and Branch‐and‐Benders‐cut algorithm are also presented. Both approaches are compared on the basis of extensive numerical experiments that bring the following conclusions. First, the survivable variant is shown to be much harder than the original Ring Star Problem, and the extra cost induced by survivability is significant. Second, the ILP formulation tends to produce tighter lower bounds but memory issues are frequent for large instances. Finally, the Branch‐and‐Benders‐cut algorithm returns feasible solutions that are often of better quality than those produced by ILP, and is less frequently subjected to memory issues on the considered set of instances.

Cost allocation in energy distribution networks

This paper presents a cost allocation problem arising from energy distribution and proposes cost allocation rules that depend on the distribution network and consumer demands. To determine relevant rules, we adopt a normative approach and compare two principles: (i) the connection principle and (ii) the uniformity principle. The Connection rule is proposed in accordance with (i), while the Uniform rule is developed in line with (ii). However, (i) and (ii) are incompatible. To make a trade-off between them, we propose a family of Mixed rules. Each rule is axiomatically characterized. Then, we demonstrate that the Connection rule coincides with the multi-choice Shapley value of a specific multi-choice game derived from the original problem. Moreover, the Connection rule is in the Core of this game. Similarly, we show that the Uniform rule and the Mixed rules coincide with other solution concepts from multi-choice games.

Shear-induced wrinkling in accelerating thin elastic discs

The wrinkling instabilities produced by in-plane angular accelerations in a rotating disc are discussed here in a particular limit of relevance to very thin plates. By coupling the classical linear elasticity solution for this configuration with the Föppl–von Kármán plate buckling equation, a fourth-order boundary-value problem with variable coefficients is obtained. The singular-perturbation character of the resulting problem arises from a combination of factors encompassing both the pre-stress (due to the spinning motion) and the geometry of the annular domain. With the help of a simplified multiple-scale perturbation method in conjunction with matched asymptotics, we succeed in capturing the dependence of the critical (wrinkling) acceleration on the instantaneous speed of the disc as well as other physical parameters. We show that the asymptotic predictions compare well with the results of direct numerical simulations of the original bifurcation problem. The limitations of the formulae obtained are also considered, and some practical suggestions for improving their accuracy are suggested.

Joint trajectory design and spectrum allocation for unmanned aerial vehicle task efficiency

AbstractIn unmanned aerial vehicle (UAV)‐assisted wireless sensor networks (WSNs), UAVs are employed to collect sensing data from each sensor node (SN). Because of limited battery capacity, shortening the time of data collection by the UAV is necessary. Notably, the task completion time is related to UAV trajectory and spectrum allocation. In this paper, joint trajectory design and spectrum allocation is studied to minimize the task completion time of UAVs while ensuring the target upload data amount for each SN. The cases that all SNs are located within the communication range of a UAV is explored first; the formulated problem is non‐convex and difficult to solve directly. Hence, an iterative algorithm based on block coordinate descent and successive convex approximation is proposed to decompose and transformed the original problem into two convex optimizations. Furthermore, the proposed algorithm is applied to the general case that all SNs are widely distributed by leveraging the spiral algorithm and traveling salesman problem technique. Simulation results show that the proposed algorithm can effectively reduce the task time of UAV compared to other benchmark algorithms.

Complexity and algorithm of setting optimal location for data sink in real-time NOMA-based IIoTs

Real-time performance is one of the most vital metrics in Non-Orthogonal Multiple Access (NOMA) based Industrial Internet of Things (IIoTs) applications. Since the relative geographic relationship between data sink and wireless sensors affects the transmission parallelism and thus the real-time performance, setting a reasonable location for the data sink is thus a feasible way to high real-time performance for applications where the locations of wireless sensors are fixed. In this paper, we consider how to find an optimal location for the data sink to minimize average access delay. We first formulate the problem and prove it to be NP-Complete by presenting an original reduction proof from classic set partition problem. Second, by tightening the NOMA decoding constraint of the original problem, a heuristic algorithm, which is designed based on Apollonian Circle Theorem and Dilworth Theorem, obtains a feasible location for the data sink. Simulation results reveal that the real-time performance loss of the heuristic algorithm is no more than 50% of the best one. Compared with the classic TDMA scheme, the real-time performance increases by more than 60% for some typical settings, and it can even reach 70% for the linear network topology, owing to the full exploitation of NOMA parallelism.

A novel spectral method and error estimates for fourth-order problems with mixed boundary conditions in a cylindrical domain

In this paper, a novel spectral method is presented and studied for the fourth order problem with mixed boundary in a cylindrical domain. The basic idea of our approach is to reduce the original problem into a series of decoupled two-dimensional fourth-order problems first, by using the cylindrical coordinate transformation and Fourier expansion, and then adopt the standard spectral method to solve the decoupled problems. A new essential pole condition is proposed to overcome the difficulty caused by the introduction of singularity and variable coefficients in cylindrical coordinate transformation. Existence and uniqueness of the weak solution and the discrete numerical solution are proved, and error estimates of the spectral method are derived. Furthermore, the efficient implementation of our algorithm is discussed, where a set of effective basis functions are constructed to ensure the sparsity of the mass matrix and stiffness matrix. Numerical examples are presented to validate the theoretical findings and the efficiency of our algorithm.