Convex Relaxation Techniques Research Articles

Control of dynamic ride-hailing networks with a mixed fleet of autonomous vehicles and for-hire human drivers

This study examines a ride-hailing platform that employs a mixed fleet of autonomous vehicles (AVs) and human drivers to offer on-demand ride-hailing services across a network over a finite planning horizon. In the mixed fleet, AVs have a fixed fleet size and are centrally managed by the platform, while human drivers can dynamically enter and exit from the ride-hailing market based on the prospective earning opportunities at various time periods. We introduce a game-theoretic model that encapsulates the strategic behavior of human drivers, passengers, and the ride-hailing platform. In this model, passengers choose between ride-hailing and alternative transportation options by considering fares and waiting times. Human drivers schedule their working hours in response to potential earnings and strategically relocate to maximize passenger pickups, while the platform adjusts pricing and AV repositioning strategies to maximize its profit. The solution is characterized as a Nash equilibrium within a two-player game, with one player representing the ride-hailing platform, and the other as a virtual representative of the collective human drivers. To address the non-convex nature of the game, we employ a convex relaxation technique to ascertain a near-optimal solution framed as an ϵ-Nash equilibrium, with ϵ accurately characterized. The proposed model and solution algorithm are validated in a case study for Manhattan, New York City. The simulation results underscore that the distribution of human drivers is closely aligned with the immediate, minute-to-minute fluctuations in passenger demand, enabling them to effectively meet the surge during peak demand windows. Conversely, the deployment patterns of AVs are more attuned to the extended, daily cycles of passenger movements throughout different city zones, from daytime to nighttime. We further compare the mixed fleet model with scenarios where the platform exclusively utilizes human drivers or AVs, respectively. The comparison show that the mixed fleet model excels in balancing the supply and demand over space and time, thereby leading to shorter wait times and reduced travel cost for ride-hailing passengers, as well as improved profit for the ride-hailing platform.

Multi-Channel Audio Completion Algorithm Based on Tensor Nuclear Norm

Multi-channel audio signals provide a better auditory sensation to the audience. However, missing data may occur in the collection, transmission, compression, or other processes of audio signals, resulting in audio quality degradation and affecting the auditory experience. As a result, the completeness of the audio signal has become a popular research topic in the field of signal processing. In this paper, the tensor nuclear norm is introduced into the audio signal completion algorithm, and the multi-channel audio signals with missing data are restored by using the completion algorithm based on the tensor nuclear norm. First of all, the multi-channel audio signals are preprocessed and are then transformed from the time domain to the frequency domain. Afterwards, the multi-channel audio with missing data is modeled to construct a third-order multi-channel audio tensor. In the next part, the tensor completion algorithm is used to complete the third-order tensor. The optimal solution of the convex optimization model of the tensor completion is obtained by using the convex relaxation technique and, ultimately, the data recovery of the multi-channel audio with data loss is accomplished. The experimental results of the tensor completion algorithm and the traditional matrix completion algorithm are compared using both objective and subjective indicators. The final result shows that the high-order tensor completion algorithm has a better completion ability and can restore the audio signal better.

Synthesizing Negative-Imaginary Systems With Closed-Loop H∞-Performance Via Static Output Feedback Control

Abstract This paper develops an algorithm for synthesizing negative imaginary (NI) systems with a specified H∞-norm bound by exploiting the convex relaxation technique in the static output-feedback (SOF) control design framework. This scheme improves the robustness objective since the present framework can handle both NI and H∞-norm-bounded unstructured uncertainty of the system. The non-strict inequality involved in the NI system description that creates a major difficulty in the SOF control design, can efficiently be tackled by the developed convex relaxation-based iterative algorithm. The advantages of the algorithm are shown using several examples and the results are compared with the existing works.

Multi-objective optimal decision for orderly power utilization based on improved ε-constraint method in active distribution networks.

With the increasing demand for electricity load in China, orderly power utilization are important measures to alleviate electricity shortages during peak periods. This article establishes a multi-objective optimization model for orderly power utilization in active distribution networks is established, with the optimization objectives of minimizing the total operation cost, minimizing the cost for users, and minimizing the load fluctuation of the system. This model contains a large number of integer variables and nonlinear constraints, which is difficult to solve. To reduce computation time, convex relaxation techniques are adopted to transform the original model into a mixed-integer second-order cone programming (MISOCP) model, which has lower computational complexity. Furthermore, The improved ε-constraint method is proposed to solve the model, which can directly and quickly find the compromise optimal solution of the multi-objective problem. By using simplex search algorithm, the proposed method dose not need to traverse all grid points, which can significantly reduce computation time. Finally, case study on the the IEEE-33 bus distribution network demonstrate the effectiveness of the proposed method.

Sensing Fairness-Based Energy Efficiency Optimization for UAV Enabled Integrated Sensing and Communication

Integrated sensing and communication (ISAC) is developing rapidly due to the advantages of bandwidth saving, hardware cost reduction and energy saving. This paper proposes a unmanned aerial vehicle (UAV) enabled ISAC (UAV-ISAC) system, where the UAV will sense ground users and forward sensed information to base station (BS). Radar mutual information (MI) is introduced to measure the UAV sensing performance. On the basis of sensing fairness for each user, a multi-objective resource optimization problem of maximizing both energy efficiency (EE) and minimum radar MI is studied by jointly optimizing user scheduling, transmit power and UAV trajectory. To solve the non-convex optimization problem, we decompose it into three sub-problems: user scheduling optimization, transmit power optimization, and UAV trajectory optimization. Each subproblem can be solved by successive convex approximation (SCA), fractional programming and relaxation technique. By iteratively optimizing the three sub-problems, we can obtain the suboptimal solution to original optimization problem. Simulation results show that the proposed optimization scheme can maximize the EE of UAV while ensuring sensing fairness for all users.

Non-Convex Optimization: Using Preconditioning Matrices for Optimally Improving Variable Bounds in Linear Relaxations

The performance of branch-and-bound algorithms for solving non-convex optimization problems greatly depends on convex relaxation techniques. They generate convex regions which are used for improving the bounds of variable domains. In particular, convex polyhedral regions can be represented by a linear system A.x=b. Then, bounds of variable domains can be improved by minimizing and maximizing variables in the linear system. Reducing or contracting optimally variable domains in linear systems, however, is an expensive task. It requires solving up to two linear programs for each variable (one for each variable bound). Suboptimal strategies, such as preconditioning, may offer satisfactory approximations of the optimal reduction at a lower cost. In non-square linear systems, a preconditioner P can be chosen such that P.A is close to a diagonal matrix. Thus, the projection of the equivalent system P.A.x=P.b over x, by using an iterative method such as Gauss–Seidel, can significantly improve the contraction. In this paper, we show how to generate an optimal preconditioner, i.e., a preconditioner that helps the Gauss–Seidel method to optimally reduce the variable domains. Despite the cost of generating the preconditioner, it can be re-used in sub-regions of the search space without losing too much effectiveness. Experimental results show that, when used for reducing domains in non-square linear systems, the approach is significantly more effective than Gauss-based elimination techniques. Finally, the approach also shows promising results when used as a component of a solver for non-convex optimization problems.

A Semidefinite Relaxation Approach for Mobile Target Localization Based on TOA and Doppler Frequency Shift Measurements

In this paper, we discuss mobile source localization using both time of arrival and Doppler frequency shift (TOA-DFS) measurements, where the source moves at a nonuniform velocity. To obtain the position of a mobile source, we first formulate the weighted least squares (WLS) problem by ignoring the second-order noise terms. Due to the non-convexity, we apply the convex relaxation technique to transform the problem into a semi-definite programming problem. However, ignoring the second-order noise terms is only reasonable in the case of small noise levels. In view of this, we then directly establish the maximum likelihood (ML) estimator based on the measurements model without ignoring the second-order noise terms. Since the ML estimator is a non-convex problem, we also propose implementable semidefinite relaxation (SDR) technique to tackle it. Finally, the Cramér-Rao lower bound (CRLB) analysis and results verify that the proposed methods based on the TOA-DFS measurements can significantly enhance localization accuracy.

Joint Power and Bandwidth Allocation in Collocated MIMO Radar Based on the Quality of Service Framework

The simultaneous multi-beam working mode of the collocated multiple-input and multiple-output (MIMO) radar enables the radar to track multiple targets simultaneously. A joint power and bandwidth allocation algorithm in a collocated MIMO radar based on the quality of service (QoS) framework is proposed for the multi-target tracking problem with different threat levels. Firstly, a posterior Cramer–Rao lower bound (PCRLB) concerning the power and bandwidth is derived. In addition, the optimal objective functions of power and bandwidth are designed based on the QoS framework, and the problem is solved using the convex relaxation technique and the cyclical minimization algorithm. The numerical results show that the proposed algorithm has better tracking accuracy and achieves more reasonable resource allocation compared to strategies such as average allocation.

Two-stage data-driven dispatch for integrated power and natural gas systems by using stochastic model predictive control

The optimal dispatch of the integrated power and natural gas systems can increase the utilization rate of renewable energy and energy efficiency while decreasing operation costs. The common prediction errors of wind power and electric load have the potential to negatively impact the normal operation of the integrated power and natural gas systems. A two-stage data-driven dispatch strategy is proposed to reduce this effect, consisting of the day-ahead dispatch stage and the intraday rolling dispatch stage using stochastic model predictive control (MPC). In the day-ahead dispatch stage, the data-driven chance constraints of tie-line power and reserve of gas-fired generators are built, and the day-ahead tie-line power is obtained and regarded as input parameters to the intraday dispatch stage. In the intraday dispatch stage, the data-driven chance constraints of tie-line power and reserve of gas-fired generators with the latest rolling prediction data are built, and the remaining control variables are obtained. The distribution characteristics of the stochastic prediction errors of wind power and electric load are captured and described by the variational Bayesian Gaussian mixture model with massive historical data. Then the original stochastic mixed-integer nonlinear programming problem is converted to a tractable deterministic one by the quantile-based analytical reformulation and convex relaxation technique. Finally, the proposed strategy is verified by the numerical experiments based on a modified IEEE 33-bus system integrated with a 10-node natural gas system and a micro hydrogen system. The numerical results demonstrate that the proposed strategy reduces the actual costs and decreases the violation rate caused by the stochastic prediction errors of wind power and electric load.

SLfRank: Shinnar-Le-Roux Pulse Design With Reduced Energy and Accurate Phase Profiles Using Rank Factorization.

The Shinnar-Le-Roux (SLR) algorithm is widely used to design frequency selective pulses with large flip angles. We improve its design process to generate pulses with lower energy (by as much as 26%) and more accurate phase profiles. Concretely, the SLR algorithm consists of two steps: (1) an invertible transform between frequency selective pulses and polynomial pairs that represent Cayley-Klein (CK) parameters and (2) the design of the CK polynomial pair to match the desired magnetization profiles. Because the CK polynomial pair is bi-linearly coupled, the original algorithm sequentially solves for each polynomial instead of jointly. This results in sub-optimal pulses. Instead, we leverage a convex relaxation technique, commonly used for low rank matrix recovery, to address the bi-linearity. Our numerical experiments show that the resulting pulses are almost always globally optimal in practice. For slice excitation, the proposed algorithm results in more accurate linear phase profiles. And in general the improved pulses have lower energy than the original SLR pulses.

Reduced-Order Static -Gain Control of Rotorcraft with Guaranteed Rotor–Fuselage Stability

A reduced-order control design method for rotorcraft is proposed in this work. Explicit constraints on the closed-loop fuselage–rotor stability are considered in the fuselage dynamics-based control design, aiming for the -gain performance for the fuselage dynamics under pole placement constraints and the guaranteed fuselage–rotor stability. Sufficient conditions for the control design problem are presented, formulated as a feasibility problem based on bilinear matrix inequalities. Such an NP-hard problem is approximated using convex relaxation techniques and the cone complementary linearization method, leading to an iterative control design algorithm. An altitude–velocity vector tracking concept is also proposed in this work, integrated with the proposed control design method to design rotorcraft trajectory controllers. Performance of the resultant rotorcraft trajectory controllers is examined via numerical simulations against representative maneuvers from the ADS-33E specification.

A novel convex relaxation technique on affine transformed sampled-data control issue for fuzzy semi-Markov jump systems

This article investigates affine transformed sampled-data control problems for fuzzy semi-Markov jump systems (FSMJSs). First of all, in the novel fuzzy sampled-data control, an affine transformed membership function is introduced, which contributes to constructing the synchronous time scale grades of membership without any constraint condition. Then, by utilizing a mode-dependent Lyapunov function with the looped functions, a sufficient condition concerning the asymptotical stability of the closed-loop FSMJSs is established in the form of linear matrix inequality (LMI). Meanwhile, to solve parameterized LMI (PLMI), a novel convex relaxation technique is proposed, based on which less conservatism stabilization criteria of FSMJSs, and a maximum sampling interval with respect to sampled-data control are further derived. Finally, two examples are carried out to manifest numerically the validity of the raised method.

Self-Calibration of Acoustic Scalar and Vector Sensor Arrays

In this work, we consider the self-calibration problem of joint calibration and direction-of-arrival (DOA) estimation using acoustic sensor arrays. Unlike many previous iterative approaches, we propose solvers that can be readily used for both linear and non-linear arrays for jointly estimating the sensor gain, phase errors, and the source DOAs. We derive these algorithms for both the conventional element-space and covariance data models. We focus on sparse and regular arrays formed using scalar sensors as well as vector sensors. The developed algorithms are obtained by transforming the underlying non-linear calibration model into a linear model, and subsequently by using convex relaxation techniques to estimate the unknown parameters. We also derive identifiability conditions for the existence of a unique solution to the self-calibration problem. To demonstrate the effectiveness of the developed techniques, numerical experiments, and comparisons to the state-of-the-art methods are provided. Finally, the results from an experiment that was performed in an anechoic chamber using an acoustic vector sensor array are presented to demonstrate the usefulness of the proposed self-calibration techniques.

Joint Strategy of Power and Bandwidth Allocation for Multiple Maneuvering Target Tracking in Cognitive MIMO Radar With Collocated Antennas

In this article, we propose a joint strategy of power and bandwidth allocation (JSPBA) for multiple maneuvering target tracking (MMTT) in the multi-input multi-output (MIMO) radar with collocated antennas. The basis of our strategy is to optimally allocate the transmitted resources of power and effective bandwidth by the prior information in the closed-loop system of cognitive tracking. On account of the predicted conditional Cramér-Rao lower bound (PC-CRLB) offering a more accurate and time-sensitive lower bound than the standard posterior CRLB (PCRLB), the PC-CRLB of the range, Doppler frequency, and direction-of-arrival (DOA) is derived, normalized and adopted as the optimization criterion. Moreover, in order to solve the nonconvex problem, the initial nonconvex problem is converted into a series of convex problems, which are further formulated as the standard semi-definite programming (SDP) problems and then be solved, by introducing the convex relaxation technique and the two-step solution technique. Simulations confirm the superiority of the proposed JSPBA algorithm, in terms of the overall tracking accuracy in the MMTT scenario.

Sensor Selection with Composite Features in Identifying User-Intended Poses for Human-Prosthetic Interfaces.

A Human-Prosthetic Interface (HPI) serves to estimate and realise the limb pose intended by the human user, using the information obtained from sensors worn by the user. In recent studies, the HPI maps multi-joint limb poses (i.e. coordinated movement of the body and limbs) to the inputs of multiple sensors. This is in contrast to the conventional methods where each degree of freedom of the powered prosthesis is mapped to the input of one/a pair of sensors. In this approach, it is necessary to systematically select sensors that carry the most information for the intended set of poses, to improve system accuracy and/or minimise the number of sensors, thus the complexity, in the prosthetic system. In this paper, sensor selection process is systematically formulated to maximise the information contained in the input features for a given number of sensors. Most importantly, it accounts for composite features, which are features requiring information from multiple sensors. Such composite features exist and are important in HPIs as we seek to capture coordinated motion involving movements of multiple limb and body segments. A non-convex optimisation problem is formulated which accounts for the constraint introduced by the composite features. A projection matrix is utilised as the optimisation variable to select intended features for evaluation. The problem is solved by the proposed Sensor Selection with Composite Features (SS-CF) algorithm which adapts convex-relaxation techniques. The SS-CF is benchmarked against HPI with expert-selected sensors in the literature and against a greedy heuristic method. The outcome demonstrated the efficacy of the SS-CF algorithm.

Distributed robust cooperative scheduling of multi-region integrated energy system considering dynamic characteristics of networks

Due to the advantages of reliability and economy, multiple adjacent region integrated energy systems (IESs) are interconnected to form a multi-region IES through multi-energy network. However, the limitation on private data exchange, multi-uncertainty and the complex dynamic characteristics bring challenges to the optimal scheduling of the multi-region IES. This paper proposes a distributed robust cooperative scheduling method for the multi-region electricity–natural gas–heat IES considering the dynamic characteristics and the uncertainties of wind turbine (WT) and electricity load. Firstly, a robust scheduling model of the multi-region IES considering dynamic characteristics and the uncertainties is built. In particular, the dynamic transmission models of gas and heating networks are constructed, and a regulation model of virtual heat storage is built to control the virtual heat storage in district heating network (DHN). Further, to preserve the information privacy and decision-making independence for each sub-region IES, a distributed cooperative scheduling framework based on the consensus-based alternating direction method of multipliers (ADMM) is developed for the multi-region IES, where the original robust scheduling model is decomposed into several subproblems that are solved independently. Moreover, the nonconvex dynamic natural gas flow function is addressed by convex relaxation technique, and the max–min robust model with binary variables is solved by alternative optimization procedure (AOP) method and duality theory. Finally, the simulation results show that the proposed method can converge reliably, realize the balance between the operation cost and the heat loss and provide a flexible scheduling scheme for the multi-region IES.

Distribution Systems AC State Estimation via Sparse AMI Data Using Graph Signal Processing

This work establishes and validates a Grid Graph Signal Processing (G-GSP) framework for estimating the state vector of a radial distribution feeder. One of the key insights from GSP is the generalization of Shannon’s sampling theorem for signals defined over the irregular support of a graph, such as the power grid. Using a GSP interpretation of Ohm’s law, we show that the system state can be well approximated with relatively few components that correspond to low-pass Graph Fourier Transform (GFT) frequencies. The target application of this theory is the formulation of a three-phase unbalanced Distribution System State Estimation (DSSE) formulation that recovers the GFT approximation of the system state vector from sparse Advanced Metering Infrastructure (AMI) measurements. To ensure convergence of G-GSP for DSSE, the proposed solution relies on a convex relaxation technique. Furthermore, we propose an optimal sensor placement algorithm for AMI measurements. Numerical results demonstrate the efficacy of the proposed method.

Optimal Communication Network Design of Microgrids Considering Cyber-Attacks and Time-Delays

Distributed secondary control stands out for its flexibility and expandability in microgrids (MGs) control, in where communication network plays a fundamental and critical role. The communication topology and link-weights have significant impact on the attack-resilience and dynamic performance of MGs, which should be appropriately designed. However, the joint optimization problem of communication topology and link-weights in the existing literature of consensus-based secondary voltage control of MGs is usually neglected. To bridge this gap, we propose a novel two-stage optimization approach for the communication network design, which jointly optimizes the topological structure to enhance the structural survivability and link weights to improve the dynamic performance (including the speed of convergence and robustness to time-delay). The first stage problem is formulated into a mixed-integer semi-define programming (MISDP) model based on convex relaxation technique, which is then converted equivalently into an integer quadratic programming (IQP) problem and then a MISDP feasibility problem to facilitate the solution. The second stage problem is formulated into a bi-objective SDP model to compromise between the convergence performance and robustness to time-delay. Simulations based on a microgrid with 10 distributed generation (DG) units under different scenarios are implemented to verify the effectiveness of the proposed method.

Sparse Array Synthesis Including Mutual Coupling for MU-MIMO Average Capacity Maximization

A hybrid optimization algorithm, including mutual coupling (MC), is proposed to synthesize an irregular sparse array (ISA) for average capacity maximization in a multiuser multiple-input–multiple-output (MU-MIMO) system. The hybrid approach is composed of two phases to suboptimally determine the location of a fixed number of omnidirectional thin dipole antennas in an arbitrary sparse aperture via a diagonal antenna selection matrix. In Phase I, the problem is relaxed to a convex optimization by ignoring the MC and weakening the constraints. The output of Phase I is accounted as a reliable initial guess for the genetic algorithm (GA) in Phase II, which incorporates the MC effects through the coupling matrix and avoids the convex relaxation technique. The proposed approach outperforms the conventional GA with a random initial population, while it avoids trying several starting positions. Meanwhile, the undesirable appearance of grating lobes, due to the undersampling, and the degrading MC effects are suppressed by aperiodicity. It is observed that doubling the conventional interelement spacing (half-wavelength) and finding the location of eight dipoles in a sparse aperture by the proposed method improve the average capacity by 3.27%–11.9% when the number of users varies from two to eight, and the signal-to-noise ratio (SNR) is 30 dB.

Double bias correction for high-dimensional sparse additive hazards regression with covariate measurement errors.

We propose an inferential procedure for additive hazards regression with high-dimensional survival data, where the covariates are prone to measurement errors. We develop a double bias correction method by first correcting the bias arising from measurement errors in covariates through an estimating function for the regression parameter. By adopting the convex relaxation technique, a regularized estimator for the regression parameter is obtained by elaborately designing a feasible loss based on the estimating function, which is solved via linear programming. Using the Neyman orthogonality, we propose an asymptotically unbiased estimator which further corrects the bias caused by the convex relaxation and regularization. We derive the convergence rate of the proposed estimator and establish the asymptotic normality for the low-dimensional parameter estimator and the linear combination thereof, accompanied with a consistent estimator for the variance. Numerical experiments are carried out on both simulated and real datasets to demonstrate the promising performance of the proposed double bias correction method.