Relaxation Algorithm Research Articles

User Pairing and Power Allocation in Untrusted Multiuser NOMA for Internet of Things

In the Internet-of-Things (IoT), massive sensitive and confidential information is transmitted wirelessly, making security a serious concern. This is particularly true when technologies, such as non-orthogonal multiple access (NOMA), are used, making it possible for users to access each other’s data. This paper studies secure communications in multiuser NOMA downlink systems, where each user is potentially an eavesdropper. Resource allocation is formulated to achieve the maximum sum secrecy rate, meanwhile satisfying the users’ data requirements and power constraint. We solve this non-trivial, mixed-integer non-linear programming problem by decomposing it into power allocation with a closed-form solution, and user pairing obtained effectively using linear programming relaxation and barrier algorithm. These subproblems are solved iteratively until convergence, with the convergence rate rigorously analyzed. Simulations demonstrate that our approach outperforms its existing alternatives significantly in the sum secrecy rate and computational complexity.

Codebook Based Antenna Configuration: A New Network Planning Paradigm for mmWave Mobile Communication Systems

To configure the azimuths and downtilts of massive antennas optimally is of great importance for the mmWave mobile communication systems to support diverse services, however, it is always computationally forbidden due to the huge number of possible combinations in real-world mobile networks. In this paper, we develop an effective and efficient method to deal with this intractable optimization task. First, we explore the most promising azimuth and downtilt configurations of each antenna with a branch-and-bound procedure followed by a semi-definite relaxation algorithm, which finally generates a codebook with finite elements to reduce the unnecessary searches for similar or unpromising configurations during global search; second, a Monte Carlo tree search method is introduced to iteratively optimize the azimuths and downtilts of all antennas, aiming at alleviating the mutual interference of adjacent antennas to maximize the signal-to-interference ratio of the target service area, during which the generated antenna configuration codebook guides the iterative search process efficiently. Experiment results in real urban scenarios show that our proposed method can produce the best signal-to-interference ratio coverage as compared with state-of-the-art ones. Moreover, the proposed algorithm can scale up straightforwardly, making it a competitive choice for large-scale network optimization.

Weighted Sum-Rate Maximization in Multi-IRS-Aided Multi-Cell mmWave Communication Systems for Suppressing ICI

Intelligent reflecting surface (IRS) has been envisioned as a disruptive technology with the capability of reconfiguring the wireless transmission environment. Hence, the IRS is used to assist the communication system, however, the impacts of inter-cell interference (ICI) on cell-edge users are severe in multi-cell communications but have been ignored in most literature about IRS-assisted communication systems. In this paper, we consider the downlink transmission of cell-edge users in multi-cell millimeter wave (mmWave) communication systems with the aid of multiple IRSs deployed at the cell boundary. To suppress the ICI, we maximize the weighted sum-rate (WSR) by jointly optimizing the IRS-user association matrix, the transmit beamforming of mmWave base stations (MBSs), and the phase shifts of IRSs, subject to the power limits of MBSs, the reflection constraints of IRSs, and the association constraints. Then, a joint IRS-user association and beamforming algorithm with alternating optimization (AO) framework is developed to tackle the mixed-integer non-convex problem. Specifically, a simplified objective function with respect to the association variables is derived and the non-convex binary constraints are handled by the relaxation operation and the first-order Taylor expansion, while the arithmetic-geometric mean inequality (AGMI)-based rank-one relaxation algorithm and the ring-based penalty convex-concave procedure (CCP) method are proposed for obtaining the transmit beamforming matrix of MBSs and the phase shifts of IRSs, respectively. Simulation results demonstrate the superiority of the proposed algorithm over the benchmark schemes. The useful insights into the optimization of IRS-user association and design of the number of elements of IRSs are also presented for future networks.

Robust multimodal hub network design for hazmat considering a detour strategy

Most traditional hub networks focus on ordinary cargo transportation and usually limit the number of transshipment occurrences to minimise costs. However, this limitation is not conducive to the transportation of hazmat, which often requires detours to avoid high risks. Therefore, this paper proposes a new multimodal network model considering a detour strategy and discusses the uncertainty in the network from the demand point of view. The method of robust optimisation is used to address the uncertainty in the proposed model, and the Lagrange relaxation algorithm and augmented ɛ-constraint method are used to transform and simplify the model. In addition, simulated data of hazmat transportation in the Yangtze River Delta urban agglomeration are used for a case study. The results show that under the consideration of uncertain demand, changes in the demand uncertainty degree and transit discount factor cause changes in the total cost and risk of the multimodal hub network, thus affecting the carrier’s location and route decisions; the new model can better control and mitigate risk, which is more suitable for hazmat transportation. This study provides a reference for multimodal transport carriers in hazmat transportation decision making.

IPRQP: a primal-dual interior-point relaxation algorithm for convex quadratic programming

IPRQP: a primal-dual interior-point relaxation algorithm for convex quadratic programming

New Heuristic Algorithm Based on the Lagrangian Relaxation for a Real-World High-Speed Railway Timetabling Problem

High-speed railway (HSR) systems have been developing rapidly in China and other countries throughout the past decade; as a result, efficiently operating these large-scale systems has presented new challenges to the railway industry. A high-quality train timetable should satisfy transportation demands with the best possible benefit. This study presents a scheduling model for double-track HSR lines based on a train service plan for timetable optimization. First, we construct a space–time network and assign each resource a usage cost at every discrete instant of time to maximize the total profit of a train timetable as a whole. Second, we propose a heuristic method based on the Lagrangian relaxation algorithm to solve the model by updating the resource usage costs according to the conflicts caused by trains in each iteration. Finally, we consider the Beijing–Shanghai HSR line as a real-world application of the methodology. In each of the four cases tested, we obtain an approximate optimal solution with a duality gap of less than 8% within 120 s. The results show that the proposed model and heuristic offer an efficient and promising means of addressing HSR timetable problems.

Lagrangean relaxation algorithms for fixed-charge capacitated relay network design

We consider a fixed-charge capacitated relay network design problem (CRNDP), which has widespread applications in the design of telecommunication and transportation networks, where relay points are presented as signal regenerators, truckload driver and/or tractor exchange stations or electrical vehicle recharging station. Our CRNDP integrates the key features of single allocation hub location problem (SAHLP) and capacitated network design problem (CNDP). Typical range restrictions in truckload transportation, electrical vehicles as well as for keeping signal strength in telecommunications, and further limitations of flow on the links of an underlying network in these mediums motivate the use of distance constraints and link capacity constraints in the design of a relay network for this diverse set of applications. To efficiently solve our model, we propose a Lagrangean relaxation algorithm, where the upper bounds are enhanced by a heuristic during the iterations and the lower bound is improved by the introduction of surrogate constraints. The extensive computational studies are conducted to examine the performance of the algorithm and the results show that the solution method we propose can efficiently solve CRNDP and obtain good-quality solutions, even for very large size instances.

A variational inequality approach for the numerical simulation of steel solidification of two-phase continuous casting process

In this paper, we study a nonlinear parabolic type model with an inequality constraint on the solution describing the phase liquid-solid heat transfer during the solidification process in continuous casting. We consider a semi-discretization with respect to time (implicit Euler method in time) of the studied thermal problem. We have a sequence of stationary constrained problems. Reformulating the problem into a variational inequality problem and thanks to the approximation (linearization) of the nonlinear boundary condition (radiation condition) and appropriate assumptions, we establish a result of existence and uniqueness of the solution of the stationary problem. We also consider a multivalued formulation of the same problem allowing the analysis of the behavior of the iterative relaxation algorithm used for the numerical solution of the discretized problem. Finally, numerical simulations are displayed.

Redesign of the sustainable wheat-flour-bread supply chain network under uncertainty: An improved robust optimization

With the passage of time, the grains supply chain may go out of optimality and be disrupted, so it is necessary to redesign the supply chain to avoid incurring irreparable costs. Because of the occurring grain supply chain sub-optimization event as time passes, this paper redesigns a wheat-flour-bread supply chain network considering sustainable development under uncertainty. For this reason, both facility and transaction networks are investigated, and newly redesigned networks are presented. First, a new definition of sustainable development pillars in supply chains is proposed, and then a four-objective mixed-integer linear programming model is presented. Due to uncertainties in the economic, environmental, social, and technical parameters of the presented model, a robust optimization method is developed to deal with them. Then, an augmented weighted Tchebycheff method, followed by the Lagrangian relaxation algorithm, is applied for solving the four-objective model. Next, a real-case study is performed to appraise the efficiency of the model. Finally, a sensitivity analysis is conducted on the critical parameters of the model to analyze the behavior of different objective functions. With the model’s solution, the redesign of the supply chain provides an enhancement in the economic, environmental, social, infrastructural, and final objective functions by about 18, 23, 31, 37, and 20%, respectively.

The time-dependent electric vehicle routing problem with drone and synchronized mobile battery swapping

Logistics enterprises are moving towards high efficiency and green. As an emerging technology, unmanned aerial vehicle (UAV, also known as drone) is very useful to cope with flexible and fast delivery. Electric vehicles (EVs), as new energy vehicles, are the technical support for green logistics development. This paper considers the last-mile delivery system in which a fleet of homogeneous EVs, equipped with a UAV each, serve a set of customers, and meanwhile battery swapping vehicles (BSVs) offer mobile battery swapping service for EVs. A mixed integer programming (MIP) model is formulated to minimize the sum of depreciation cost, driver wages and energy consumption cost of EVs, UAVs, and BSVs, by considering the nonlinear energy consumption of each vehicle, time-dependent speeds of EVs and BSVs and monetary time value on depreciation cost. A large neighborhood search algorithm based on Q-learning (LNS-QL) is designed in which Q-learning is used to select a combination of tailored destroy and repair operators to solve three sub-problems, (1) the UAV multi-visit problem, (2) the EV and UAV synchronized scheduling problem, (3) the EV and BSV synchronized mobile battery swapping problem. Besides, this paper also designs Lagrangian relaxation algorithm for comparison with LNS-QL. Extensive experiments verify the correctness of the model by testing small-scale instances. Further analysis demonstrates the accuracy and stability of LNS-QL on medium and large-scale instances as well as computational time and verifies the effectiveness of the mobile battery swapping mode in the cooperative delivery system of EVs and UAVs. Actual cases analyze the characteristics of EV delivery from the cost and emissions perspectives by comparing with FV delivery.

Why Fourier mode analysis in time is different when studying Schwarz Waveform Relaxation

Schwarz waveform relaxation methods are Schwarz methods applied to evolution problems. Like for steady problems, they are based on an overlapping domain decomposition of the spatial domain, and an iteration which only requires subdomain solutions, now in space-time, to get better and better approximations of the global, monodomain solution. Fourier analysis has been used to study the convergence of both Schwarz and Schwarz waveform relaxation methods. We show here that their convergence is however quite different: for steady problems of diffusive type, Schwarz methods converge linearly, which is also well predicted by Fourier analysis. For a time dependent heat equation however, the Schwarz waveform relaxation algorithm first has a rapid convergence phase, followed by a slow down, and eventually convergence increases again to become superlinear, none of which is predicted by classical Fourier analysis. Introducing a new Fourier analysis combined with kernel estimates, we can explain this behavior for the heat equation. We then generalize our approach to the case of advection reaction diffusion problems. We illustrate all our results with numerical experiments.

Multidimensional rank-one convexification of incremental damage models at finite strains

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, our experiments showed that the relaxation by rank-one convexification delivering an approximation to the quasiconvex envelope prevents mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.

P2P transaction method for distributed energy prosumers based on reputation value

Adding a reputation incentive system to peer-to-peer (P2P) energy transactions can encourage prosumers to regulate their trading behavior, which is important for ensuring the efficiency and reliability of P2P transactions. This study proposed a P2P transaction mechanism and game optimization model for prosumers involved in distributed energy sources considering reputation-value incentives. First, the deviation of P2P transactions and the non-consumption rate of distributed renewable energy in P2P transactions were established as indicators to quantify the influencing factors of the reputation value, and a reputation incentive model of P2P transactions for prosumers was constructed. Then, the penalty coefficient was applied to the cost function of the prosumers, and a non-cooperative game model of P2P transactions based on the complete information of multi-prosumers was established. Furthermore, the Nash equilibrium problem was transformed into a nonlinear optimization problem by constructing the modified optimal reaction function, and the Nash equilibrium solution of the game was obtained via a relaxation algorithm. Finally, the modified IEEE 33-node test system based on electricity market P2P and an IEEE 123-node test system were used to analyze and verify the cost and P2P participation of prosumers considering the reputation value. The results show that the addition of the reputation incentive system can encourage prosumers to standardize their interactive transaction behavior and actively participate in P2P transactions. It can also improve the operation efficiency of the power grid and promote the perfection of the P2P transaction mechanism.

Relaxation-Based Radiometric Normalization for Multitemporal Cross-Sensor Satellite Images

Multitemporal cross-sensor imagery is fundamental for the monitoring of the Earth’s surface over time. However, these data often lack visual consistency because of variations in the atmospheric and surface conditions, making it challenging to compare and analyze images. Various image-normalization methods have been proposed to address this issue, such as histogram matching and linear regression using iteratively reweighted multivariate alteration detection (IR-MAD). However, these methods have limitations in their ability to maintain important features and their requirement of reference images, which may not be available or may not adequately represent the target images. To overcome these limitations, a relaxation-based algorithm for satellite-image normalization is proposed. The algorithm iteratively adjusts the radiometric values of images by updating the normalization parameters (slope (α) and intercept (β)) until a desired level of consistency is reached. This method was tested on multitemporal cross-sensor-image datasets and showed significant improvements in radiometric consistency compared to other methods. The proposed relaxation algorithm outperformed IR-MAD and the original images in reducing radiometric inconsistencies, maintaining important features, and improving the accuracy (MAE = 2.3; RMSE = 2.8) and consistency of the surface-reflectance values (R2 = 87.56%; Euclidean distance = 2.11; spectral angle mapper = 12.60).

Multiobjective Optimization for Adaptive Offloading in Distributed Multiuser MIMO Cell-Free 6G Networks

By offloading some computational tasks to the edge server, edge computing can help relieve the increasing computation burden of mobile users and improve the quality of experience of users. In this paper, we investigate the computation offloading in edge computing-enabled multiuser cell-free (CF) multi-input multi-output (MIMO) networks with multiple building baseband units (BBUs). The user association, transmit covariances, CPU-cycle frequencies, and allocated bandwidths are jointly optimized to minimize two objectives: the energy consumption of mobile devices (MDs) and execution delay of tasks. We formulate it as a vector optimization problem and adopt the scalarization technique to transform it into a scalar mixed integer nonlinear programming (MINLP). Then, to solve the MINLP, we introduce user sorting into the framework of branch and bound and propose a sorted MD association (MDA) method. Two key issues are tackled in the proposed MDA, i.e., the lower bound of the MINLP and the updation of incumbent solution. For the first issue, we present a Lagrangian dual relaxation algorithm based on subgradient projection, while the second one involving another nonconvex minimization problem, we propose a successive convex approximation (SCA) based algorithm to solve it. Finally, extensive simulations demonstrate that the proposed method is able to reduce the system cost significantly and achieve better system performance by comparing with the benchmark schemes.

Joint Optimization of Trajectory and Discrete Reflection Coefficients for UAV-Aided Backscatter Communication System with NOMA

Backscatter communication is a promising technology for the Internet of Things (IoT) systems with low-energy consumption, in which the data transmission of the backscatter devices relies on reflecting the incident signal. However, limited by the low power characteristic of the reflected signal from backscatter devices, achieving efficient data collection for the widely distributed backscatter devices is a thorny problem. Considering that unmanned aerial vehicles (UAVs) have flexible deployment capability, employing UAVs in a backscatter communication network can achieve feasible data collection for backscatter devices. In this paper, we consider a UAV-aided backscatter system and introduce Non-orthogonal multiple access (NOMA) to enable the UAV to collect signals from multiple backscatter devices simultaneously. We formulate an optimization problem to maximize the communication throughput of the considered system by jointly designing the backscatter device matching, the trajectory of the UAV, and the reflection coefficients of the backscatter devices, which is a non-convex optimization problem and challenging to solve. Hence, we decouple the original problem into three sub-problems and propose an efficient iterative algorithm based on Block Coordinate Descent (BCD) to solve them. In detail, a game-based matching algorithm is designed to ensure the transmission needs of remote backscatter devices. The UAV trajectory and reflection coefficients of backscatter devices are optimized through the Successive Convex Approximation (SCA) algorithm and relaxation algorithm. By iterative optimization of the sub-problems, the original problem is solved. The simulation results show that the proposed scheme can obtain a significant throughput gain compared to benchmark schemes.

Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p-center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented.

A unified feature-spatial cycle consistency fusion framework for robust image matching

Robust image matching is a fundamental and long-standing open problem in computer vision. Conventional wisdom has exploited redundancy to improve the robustness of image matching (e.g., from pairwise to multi-image correspondence), which works well in the spatial domain. Inspired by the success of global optimization-based approaches, we propose a novel extension of cycle consistency from multi-image to multi-descriptor matching in this paper, which integrates useful information from the feature domain. More specifically, we build upon previous work of permutation synchronization and construct a novel cycle consistency model for multi-descriptor matching. The construction of cycle consistency model is based on the analogy between multi-image matching and multi-descriptor matching in a virtual universe. It allows us to formulate multi-image and multi-descriptor matching as a constrained global optimization problem. We have developed a spectral relaxation algorithm to solve this optimization problem, admitting an efficient implementation via fast singular value decomposition (SVD). To demonstrate the robustness of the proposed method named Cycle Consistency Fusion (C2F), we have evaluated it in terms of both raw matching accuracy (pairwise or multi-image) and several higher level downstream tasks such as homography and camera pose estimation. Extensive experimental results have shown that our C2F outperforms state-of-the-art methods consistently across different datasets and vision tasks.

Millimeter-Wave Radar Monitoring for Elder’s Fall Based on Multi-View Parameter Fusion Estimation and Recognition

Human activity recognition plays a vital role in many applications, such as body falling surveillance and healthcare for elder’s in-home monitoring. Instead of using traditional micro-Doppler signals based on time-frequency distribution, we turn to another way and use the Relax algorithm to process the radar echo so as to obtain the required parameters. In this paper, we aim at the multi-view idea in which two radars at different views work synchronously and fuse the features extracted from each radar, respectively. Furthermore, we discuss the common estimated time-frequency features and time-varying spatial features of multi-view radar-echo and then formulate the parameters matrix via principal component analysis, and finally transform them into the machine learning classifiers to make further comparisons. Simulations and results show that our proposed multi-view parameter fusion idea could lead to relative-high accuracy and robust recognition performance, which would provide a feasible application for future human–computer monitoring scenarios.

An improved relaxed inertial projection algorithm for solving the minimum-norm solution of variational inequality and fixed point problems

An improved relaxed inertial projection algorithm for solving the minimum-norm solution of variational inequality and fixed point problems