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This paper presents two novel algorithmic frameworks to address the logit-based stochastic user equilibrium traffic assignment problem (SUE-TAP). Following the different variant of the gradient projection (termed as GP2) algorithm, we propose an improved GP2 algorithm (IGP) for the SUE-TAP. This study initially presents a smart approach for determining the allocation of more or less effort to specific origin–destination (OD) pairs. Subsequently, the TAP can be decomposed by different OD pairs, whereas the proposed IGP algorithm is designed based on the serial scheme (i.e., the Gauss–Seidel method). Therefore, a new parallel algorithm P-IGP is proposed, which integrates the block coordinate descent (BCD) method and the IGP algorithm. In specific, the independent OD pairs can be separated into several blocks, and the OD-based restricted subproblems within each block can be solved in parallel. Then, we outline the entire process of implementing the P-IGP algorithm to address the SUE-TAP. Several numerical experiments are conducted to verify the proposed algorithms. The results reveal that the proposed IGP algorithm demonstrates significantly speeder convergence in comparison to the traditional GP2 algorithm, achieving a remarkable acceleration of approximately 12%. Furthermore, the performance of the P-IGP algorithm surpasses that of the proposed IGP algorithm, and it can further achieve a notable 4–5-fold enhancement in convergence efficiency. [ABSTRACT FROM AUTHOR]

Summary: In postdisaster rescue scenario, unmanned aerial vehicles (UAVs) are effective tools to help ground users in disaster areas to transmit rescue‐critical data to the relief center in time due to their flexible mobilities and fast deployment. However, how UAVs choose disaster areas to take part in traffic offloading with limited bandwidth resources is challenging. This article investigates UAVs' access selection and disaster areas' bandwidth allocation scheme by jointly optimizing network throughput, bandwidth and UAVs' energy cost. The UAVs form coalitions to participate in traffic offloading cooperatively, where each UAV selects a disaster area independently. Specifically, the bandwidth resources are dynamically allocated to all disaster areas when a new coalition partition forms. The UAVs adjust their access choices for disaster areas when the bandwidth allocation changes. When no UAVs migrate to other disaster areas, the network achieves a stable state. The UAVs fly to the final selected disaster area and provide traffic offloading services. To resolve traffic offloading competitions yet enable cooperations among UAVs, we address the UAVs' access selection issue by coalition formation game. A gradient projection method is then proposed to allocate bandwidth resources which maximizes the benefit of the network. We demonstrate that the UAVs' access selection and disaster areas' bandwidth allocation algorithms are convergent. The simulation results demonstrate that our proposed cooperation order is better than the pareto and selfish orders, thereby increasing the benefit of the network. [ABSTRACT FROM AUTHOR]

One of the classical problems of optimization theory in a finite-dimensional space is to find a minimum of a function on a nonempty set. Usually, finding the precise solution to this task analytically requires a lot of computational resources or is even impossible at all. So, approximate methods are used most often in practical cases. One of the simplest and the most well-known among such approximate methods for unconditional optimization is the method of gradient descent; its generalization for conditional optimization was found in 1964, the method of projected gradient. For some simple sets (line segment, parallelepiped, ball), the projection of the point on the set can be easily found by an explicit formula. However, for more complicated sets (e.g., an ellipse), projecting becomes a separate task. Nevertheless, sometimes computing projection can be simplified by affine transform; e.g., an ellipse can be transformed into a ball by affine (moreover, by linear) transformation. The paper aims to simplify the problem of minimizing function on the set by changing the condition set by affine transform F(x)= Ax+b, where A is a non-degenerated square matrix, and b is a fixed vector of proper dimension. [ABSTRACT FROM AUTHOR]

A multilevel Monte Carlo (MLMC) method is applied to simulate a stochastic optimal problem based on the gradient projection method. In the numerical simulation of the stochastic optimal control problem, the approximation of expected value is involved, and the MLMC method is used to address it. The computational cost of the MLMC method and the convergence analysis of the MLMC gradient projection algorithm are presented. Two numerical examples are carried out to verify the effectiveness of our method.

Continual learning aims to avoid catastrophic forgetting and effectively leverage learned experiences to master new knowledge. Existing gradient projection approaches impose hard constraints on the optimization space for new tasks to minimize interference, which simultaneously hinders forward knowledge transfer. To address this issue, recent methods reuse frozen parameters with a growing network, resulting in high computational costs. Thus, it remains a challenge whether we can improve forward knowledge transfer for gradient projection approaches using a fixed network architecture. In this work, we propose the Restricted Orthogonal Gradient prOjection (ROGO) framework. The basic idea is to adopt a restricted orthogonal constraint allowing parameters optimized in the direction oblique to the whole frozen space to facilitate forward knowledge transfer while consolidating previous knowledge. Our framework requires neither data buffers nor extra parameters. Extensive experiments have demonstrated the superiority of our framework over several strong baselines. We also provide theoretical guarantees for our relaxing strategy.

A multilevel Monte Carlo (MLMC) method is applied to simulate a stochastic optimal problem based on the gradient projection method. In the numerical simulation of the stochastic optimal control problem, the approximation of expected value is involved, and the MLMC method is used to address it. The computational cost of the MLMC method and the convergence analysis of the MLMC gradient projection algorithm are presented. Two numerical examples are carried out to verify the effectiveness of our method. [ABSTRACT FROM AUTHOR]

Early time series classification (ETSC) is crucial for real-world time-sensitive applications. This task aims to classify time series data with least timestamps at the desired accuracy. Early methods used fixed-length time series to train the deep models, and then quit the classification process by setting specific exiting rules. However, these methods may not adapt to the length variation of flow data in ETSC. Recent advances have proposed end-to-end frameworks, which leveraged the Recurrent Neural Networks to handle the varied-length problems, and the exiting subnets for early quitting. Unfortunately, the conflict between the classification and early exiting objectives is not fully considered. To handle these problems, we decouple the ETSC task into the varied-length TSC task and the early exiting task. First, to enhance the adaptive capacity of classification subnets to the data length variation, a feature augmentation module based on random length truncation is proposed. Then, to handle the conflict between classification and early exiting, the gradients of these two tasks are projected into a unified direction. Experimental results on 12 public datasets demonstrate the promising performance of our proposed method. [ABSTRACT FROM AUTHOR]

Network capacity, defined as the largest sum of origin–destination (O–D) flows that can be accommodated by the network based on link performance function and traffic equilibrium assignment, is a critical indicator of network-wide performance assessment in transportation planning and management. The typical modeling rationale of estimating network capacity is to formulate it as a mathematical programming (MP), and there are two main approaches: single-level MP formulation and bi-level programming (BLP) formulation. Although single-level MP is readily solvable, it treats the transportation network as a physical network without considering level of service (LOS). Albeit BLP explicitly models the capacity and link LOS, solving BLP in large-scale networks is challenging due to its non-convexity. Moreover, the inconsideration of trip LOS makes the existing models difficult to differentiate network capacity under various traffic states and to capture the impact of emerging trip-oriented technologies. Therefore, this paper proposes the α-max capacity model to estimate the maximum network capacity under trip or O–D LOS requirement α. The proposed model improves the existing models on three aspects: (a) it considers trip LOS, which can flexibly estimate the network capacity ranging from zero to the physical capacity including reserve, practical and ultimate capacities; (b) trip LOS can intuitively reflect users' maximum acceptable O–D travel time or planners' requirement of O–D travel time; and (c) it is a convex and tractable single-level MP. For practical use, we develop a modified gradient projection solution algorithm with soft constraint technique, and provide methods to obtain discrete trip LOS and network capacity under representative traffic states. Numerical examples are presented to demonstrate the features of the proposed model as well as the solution algorithm. [ABSTRACT FROM AUTHOR]

In this treatise, we propose to design a transmit precoder at the base station (BS) for a hybrid fully connected sub-array (FCSA) architecture based on gradient projection (GP) and alternating minimization techniques. A random complex multiplicative factor comprehensively models the non-ideal nature of the power amplifier (PA), data converters (ADCs/DACs), and phase shifters (PSs) used in the transmitter radio frequency (RF) chain at the BS with the ideal mmWave channel. An effective noise term in the received signal with impairments is introduced, and the power of the same is derived. The effect of scattering of mmWave due to rainfall is taken into consideration. Two separate cases with the above-discussed impairments of (A) single user (SU) multiple-input multiple-output (MIMO) system and (B) multi-user MIMO (MU-MIMO) system are conceived. Both perfect and imperfect knowledge of the scattering parameters is considered. A linear minimum mean square error (LMMSE) channel estimation with the above impairments at the BS in the presence of rainfall scattering is performed. For both SU-MIMO and MU-MIMO systems, the spectral efficiency (SE) is evaluated for different rainfall rates at two carrier frequencies, viz. 38 GHz and 75 GHz. The channel estimate’s mean square error (MSE) for ideal and hardware impaired systems is evaluated at various rainfall rates. The performance of channel estimation is evaluated with numerical simulation. [ABSTRACT FROM AUTHOR]

In comparison to conventional human-driven vehicles (HVs), connected and automated vehicles (CAVs) provide benefits (e.g., reducing travel time and improving safety). However, before the period of fully CAVs appears, there will be a situation in which both HVs and CAVs are present, and the traffic flow pattern may differ from that of a single class (e.g., HV or CAV). In this study, we developed a multi-class traffic assignment problem (TAP) for a transportation network that explicitly considered mixed modes (e.g., HV and CAV). As a link's travel time is dependent on the degree of mixed flows, each mode required an asymmetric interaction cost function. For TAP, the multi-class user equilibrium (UE) model was used for the route choice model. A route-based variational inequality (VI) formulation was used to represent the multi-class TAP and solve it using the gradient projection (GP) algorithm. It has been demonstrated that the GP algorithm is an effective route-based solution for solving the single-class user equilibrium (UE) problem. However, it has rarely been applied to solving asymmetric UE problems. In this study, the single-class GP algorithm was extended to solve the multi-class TAP. The numerical results indicated the model's efficacy in capturing the features of the proposed TAP utilizing a set of simple networks and real transportation networks. Additionally, it demonstrated the computational effectiveness of the GP algorithm in solving the multi-class TAP. [ABSTRACT FROM AUTHOR]

Novel sparse reconstruction algorithms are proposed for beamspace channel estimation in massive multiple-input multiple-output systems. The proposed algorithms minimize a least-squares objective having a nonconvex regularizer. This regularizer removes the penalties on a few large-magnitude elements from the conventional $\ell _{1}$ -norm regularizer, and thus it only forces penalties on the remaining elements that are expected to be zeros. Accurate and fast reconstructions can be achieved by performing gradient projection updates within the framework of difference of convex functions (DC) programming. A double-loop algorithm and a single-loop algorithm are proposed via different DC decompositions, and these two algorithms have distinct computational complexities and convergence rates. An extension algorithm is further proposed by designing new step sizes for the single-loop algorithm. The extension algorithm has a faster convergence rate and can achieve approximately the same level of accuracy as the proposed double-loop algorithm. Numerical results show significant advantages of the proposed algorithms over existing reconstruction algorithms in terms of reconstruction accuracies and runtimes. Compared with the benchmark channel estimation approaches, the proposed algorithms can achieve smaller channel reconstruction error and higher achievable spectral efficiency. [ABSTRACT FROM AUTHOR]

• Network capacity estimation problem with second-best constraints (NCSC). • Systematic modelling framework for the NCSC based on bi-level model. • Modified improved gradient projection (MIGP) algorithm for combined modal split and traffic assignment. • The capacity estimation method based on sensitivity analysis. Transport network capacity enhancement is a significant aspect of urban transport planning and demand management, and a suitable measurement of the network capacity is of considerable importance. In this paper, the network capacity with second-best constraints (NCSC) is investigated to meet some specific development requirements of urban transport networks. Herein, the network capacity is restricted to an inferior "second-best solution", due to various concerns/constraints regarding the public transport mode share, serviceability, and emissions, etc. For the sake of presentation, these constraints are termed as second-best constraints, and the NCSC problem can also be referred as second-best network capacity (SBNC) problem. A bi-level model is formulated to analyse the NCSC problem. The upper-level model maximizes the total origin-destination (OD) demand, which incorporates the second-best constraints into consideration. The lower-level model is a transport network equilibrium model, which measures the network performance under a given OD demand pattern. To better investigate some important second-best constraints (e.g., public transport mode share) and also the demand elasticity, the modelling framework is extended to a multimodal transport network. An exact solution method is developed for the NCSC problem; wherein, a modified improved gradient projection (MIGP) algorithm is designed for the lower-level multimodal flow equilibrium problem, and a tailored sensitivity analysis-based (SAB) method is employed for solving the NCSC problem. The proposed models and solution methods are verified by numerical examples, demonstrating that NCSC can be an efficient tool for transport planning and management. [ABSTRACT FROM AUTHOR]

Phase-only beamforming (POB) plays an important role in modern radar and communication systems. The NP-hard nature of POB makes it difficult to be solved for large-scale antenna arrays. This paper studies two fast and simple GP algorithms for the problem of POB. First we give a new interpretation of the GP method under the framework of sequential quadratic programming (SQP). Then we prove that the GP operation gives a strict global minimizer of the subproblem of SQP. Moreover, the larger the step size in GP operation, the smaller the minimum value of the subproblem. Based on this observation we design two variable step size GP (VSGP) algorithms that improve the convergence speed of the GP method significantly. Furthermore, the VSGP algorithms have a simple structure, making them easy to be implemented in chips. Finally, we derive a necessary and sufficient condition for the Karush-Kuhn-Tucker (KKT) points of the POB problem. We also prove that, for any initialization, all limit points of the iterates generated by the VSGP algorithms converge to the KKT points. Simulation results show that the proposed algorithms outperform the state-of-the-art algorithms in terms of convergence speed and computational cost. [ABSTRACT FROM AUTHOR]

Early time series classification (ETSC) is crucial for real-world time-sensitive applications. This task aims to classify time series data with least timestamps at the desired accuracy. Early methods used fixed-length time series to train the deep models, and then quit the classification process by setting specific exiting rules. However, these methods may not adapt to the length variation of flow data in ETSC. Recent advances have proposed end-to-end frameworks, which leveraged the Recurrent Neural Networks to handle the varied-length problems, and the exiting subnets for early quitting. Unfortunately, the conflict between the classification and early exiting objectives is not fully considered. To handle these problems, we decouple the ETSC task into the varied-length TSC task and the early exiting task. First, to enhance the adaptive capacity of classification subnets to the data length variation, a feature augmentation module based on random length truncation is proposed. Then, to handle the conflict between classification and early exiting, the gradients of these two tasks are projected into a unified direction. Experimental results on 12 public datasets demonstrate the promising performance of our proposed method.

Generalized fuzzy c-means (GFCM) is an extension of fuzzy c-means using L p -norm distances. However, existing methods cannot solve GFCM with m = 1. To solve this problem, we define a new kind of clustering models, called L p -norm probabilistic K-means ( L p -PKM). Theoretically, L p -PKM is equivalent to GFCM at m = 1, and can have nonlinear programming solutions based on an efficient active gradient projection (AGP) method, namely, inverse recursion maximum-step active gradient projection (IRMSAGP). On synthetic and UCI datasets, experimental results show that L p -PKM performs better than GFCM (m > 1) in terms of initialization robustness, p-influence, and clustering performance, and the proposed IRMSAGP also achieves better performance than the traditional AGP in terms of convergence speed. [ABSTRACT FROM AUTHOR]

In this paper, we study the elastic-demand bi-criteria traffic assignment problem under the continuously distributed value of time, referred to as the ED-CBiTA problem for simplicity. Specifically, the origin and destination (O–D) demand of ED-CBiTA is endogenously guided by the expected generalized travel time aggregated from all efficient paths, and the consideration of user heterogeneity regarding the tradeoff between time and toll is accomplished by incorporating a continuously distributed value of time. We present a variable demand formulation and an equivalent excess demand reformulation for the ED-CBiTA problem. Based on two types of Gauss–Seidel decomposition schemes, we propose a novel two-stage gradient projection (TSGP) algorithm, which implicitly delivers visual interpretations to depict the interplay of supply and demand interactions. The first stage, called demand equilibration, aims to adjust O–D demand and all efficient path flows "vertically upward or downward" based on the level of network congestion. The second stage, namely boundary equilibration, is to perform the boundary movements and adjust adjacent efficient flows "horizontally forward or backward", to achieve exact positions along the Pareto frontier. Numerical results on a small network show TSGP's features and confirm that TSGP significantly outperforms two link-based benchmark algorithms. For instances of practical network size, TSGP consistently promises to obtain high-quality solutions with a rather smaller CPU time. • Propose a novel two-stage gradient projection algorithm for the ED-CBiTA problem. • The first stage of TSGP adjusts path flows vertically upwards or downwards. • The second stage of TSGP adjusts path flows horizontally forwards or backwards. • TSGP outperforms two existing benchmark algorithms in large-scale networks. [ABSTRACT FROM AUTHOR]

In this letter, we propose a novel constant envelope precoding (CEP) design for multi-user multi-input single-output (MU-MISO) downlink, where we exploit the extra degrees of freedom (DoFs) by allowing individual rotation and scaling for each constellation point. By jointly optimizing the transmitting signal matrix and the complex scaling factors, the signal-to-interference-and-noise-ratio (SINR) at the receiver is maximized. The formulated optimization problem is divided into two sub-problems and solved by the alternating optimization approach. We further propose a simplified gradient descent algorithm and extend our algorithm to frame-level precoding design. Simulation results show that the proposed algorithms outperform the existing CEP algorithms in terms of both symbol error rate (SER) and computational complexity. [ABSTRACT FROM AUTHOR]

The sparse inverse covariance matrix is used to model conditional dependencies between variables in a graphical model to fit a multivariate Gaussian distribution. Estimating the matrix from data are well known to be computationally expensive for large-scale problems. Sparsity is employed to handle noise in the data and to promote interpretability of a learning model. Although the use of a convex ℓ1 regularizer to encourage sparsity is common practice, the combinatorial ℓ0 penalty often has more favorable statistical properties. In this paper, we directly constrain sparsity by specifying a maximally allowable number of nonzeros, in other words, by imposing an ℓ0 constraint. We introduce an efficient approximate Newton algorithm using warm starts for solving the nonconvex ℓ0-constrained inverse covariance learning problem. Numerical experiments on standard data sets show that the performance of the proposed algorithm is competitive with state-of-the-art methods. Summary of Contribution: The inverse covariance estimation problem underpins many domains, including statistics, operations research, and machine learning. We propose a scalable optimization algorithm for solving the nonconvex ℓ0-constrained problem. [ABSTRACT FROM AUTHOR]

In this letter, we propose an optimal voltage feedback control algorithm for wind farms where the voltage set-point of wind-farm-side voltage source converter (WFVSC) and reactive powers of wind turbines (WTs) are coordinated. The proposed control establishes on the gradient descent and operates in an online fashion. Each WT updates its reactive power based on the local instantaneous terminal voltage and therefore, the proposed control can be implemented in a decentralized manner. Besides, we also address its asynchronous counterpart to enhance the practical applicability in real world. [ABSTRACT FROM AUTHOR]

In comparison to conventional human-driven vehicles (HVs), connected and automated vehicles (CAVs) provide benefits (e.g., reducing travel time and improving safety). However, before the period of fully CAVs appears, there will be a situation in which both HVs and CAVs are present, and the traffic flow pattern may differ from that of a single class (e.g., HV or CAV). In this study, we developed a multi-class traffic assignment problem (TAP) for a transportation network that explicitly considered mixed modes (e.g., HV and CAV). As a link’s travel time is dependent on the degree of mixed flows, each mode required an asymmetric interaction cost function. For TAP, the multi-class user equilibrium (UE) model was used for the route choice model. A route-based variational inequality (VI) formulation was used to represent the multi-class TAP and solve it using the gradient projection (GP) algorithm. It has been demonstrated that the GP algorithm is an effective route-based solution for solving the single-class user equilibrium (UE) problem. However, it has rarely been applied to solving asymmetric UE problems. In this study, the single-class GP algorithm was extended to solve the multi-class TAP. The numerical results indicated the model’s efficacy in capturing the features of the proposed TAP utilizing a set of simple networks and real transportation networks. Additionally, it demonstrated the computational effectiveness of the GP algorithm in solving the multi-class TAP.

Acting as a key to future environmentally friendly transportation systems, electric vehicles (EVs) have attached importance to develop fast charging technologies to accomplish the requirement of vehicle users. However, fast charging behaviors would cause degradations in EVs’ batteries, as well as negative effects like new demand peak and feeder overloads to the connected distribution network, especially when plugging in large scale EVs. Decentralized coordination is encouraged and our goal is to achieve an optimal strategy profile for EVs in a decentralized way considering both the need of fast charging and reducing degradations in batteries and the distribution network. In this article, we innovatively model the EV fast charging problem as an optimization coordination problem subject to the coupled feeder capacity constraints in the distribution network. The need of fast charging is expressed by the total charging time, and the relative tendency to fully charge within the desired time period. We introduce a $\ell _{0}$ -norm of the charging strategy which is non-convex to represent the total charging time, and apply the $\ell _{1}$ -norm minimization to approximate the sparse solution of $\ell _{0}$ -norm minimization. The shorter the charging horizon is the stronger willing of fast charging the user has. The objective of the optimization problem tradeoffs the EVs’ battery degradation cost, the load regulation in the distribution network, the satisfaction of charging and the total charging time, which is non-separable among individual charging behaviors. Even though alternating direction method of multipliers (ADMM) has been widely applied in distributed optimization with separable objective and coupled constraints, its decentralized scheme cannot be applied directly to the underlying non-separable EV charging coordination problem. Hence, a hierarchical algorithm based on ADMM is proposed such that the convergence to the optimal strategies is guaranteed under certain step-size parameter. Furthermore, a receding horizon based algorithm is proposed considering the forecast errors on the base demand and the EV arrival distribution. The results are demonstrated via some simulation results. [ABSTRACT FROM AUTHOR]

Purpose: There are a number of algorithms for smooth l 0 -norm (SL0) approximation. In most of the cases, sparsity level of the reconstructed signal is controlled by using a decreasing sequence of the modulation parameter values. However, predefined decreasing sequences of the modulation parameter values cannot produce optimal sparsity or best reconstruction performance, because the best choice of the parameter values is often data-dependent and dynamically changes in each iteration., Approach: We propose an adaptive compressed sensing magnetic resonance image reconstruction using the SL0 approximation method. The SL0 approach typically involves one-step gradient descent of the SL0 approximating function parameterized with a modulation parameter, followed by a projection step onto the feasible solution set. Since the best choice of the parameter values is often data-dependent and dynamically changes in each iteration, it is preferable to adaptively control the rate of decrease of the parameter values. In order to achieve this, we solve two subproblems in an alternating manner. One is a sparse regularization-based subproblem, which is solved with a precomputed value of the parameter, and the second subproblem is the estimation of the parameter itself using a root finding technique., Results: The advantage of this approach in terms of speed and accuracy is illustrated using a compressed sensing magnetic resonance image reconstruction problem and compared with constant scale factor continuation based SL0-norm and adaptive continuation based l 1 -norm minimization approaches. The proposed adaptive estimation is found to be at least twofold faster than automated parameter estimation based iterative shrinkage-thresholding algorithm in terms of CPU time, on an average improvement of reconstruction performance 15% in terms of normalized mean squared error., Conclusions: An adaptive continuation-based SL0 algorithm is presented, with a potential application to compressed sensing (CS)-based MR image reconstruction. It is a data-dependent adaptive continuation method and eliminates the problem of searching for appropriate constant scale factor values to be used in the CS reconstruction of different types of MRI data., (© 2024 Society of Photo-Optical Instrumentation Engineers (SPIE).)

This paper presents on four chosen benchmarks an experimental evidence of efficiency of active-set-based algorithms and a gradient projection scheme exploiting Barzilai–Borwein-based steplength rule for box-constrained quadratic programming problems, which have theoretically proven rate of convergence. The crucial phase of active-set-based algorithms is the identification of the appropriate active set combining three types of steps—a classical minimization step, a step expanding the active set and a step reducing it. Presented algorithms employ various strategies using the components of the gradient for an update of this active set to be fast, reliable and avoiding undesirable oscillations of active set size. [ABSTRACT FROM AUTHOR]

Summary: This paper is concerned with the proportional fairness (PF) of the spectral efficiency (SE) maximization of uplinks in a cell‐free (CF) massive multiple‐input multiple‐output (MIMO) system in which a large number of single‐antenna access points (APs) connected to a central processing unit (CPU) serve many single‐antenna users. To detect the user signals, the APs use matched filters based on the local channel state information while the CPU deploys receiver filters based on knowledge of channel statistics. We devise the maximization problem of the SE PF, which maximizes the sum of the logarithm of the achievable user rates, as a jointly nonconvex optimization problem of receiver filter coefficients and user power allocation subject to user power constraints. To handle the challenges associated with the nonconvexity of the formulated design problem, we develop an iterative algorithm by alternatively finding optimal filter coefficients at the CPU and transmit powers at the users. While the filter coefficient design is formulated as a generalized eigenvalue problem, the power allocation problem is addressed by a gradient projection (GP) approach. Simulation results show that the SE PF maximization not only offers approximately the achievable sum rates as compared to the sum‐rate maximization but also provides an improved trade‐off between the user rate fairness and the achievable sum rate. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm by incorporating inertial terms in a subgradient extragradient algorithm for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The strong convergence of the algorithm is obtained under mild assumptions. We also provide some numerical examples to illustrate that the acceleration of our algorithm is effective. [ABSTRACT FROM AUTHOR]

The semi-smooth Newton method for solving discretized contact problems with Tresca friction in three-dimensional space is analysed. The slanting function is approximated to get symmetric inner linear systems. The primal–dual algorithm is transformed into the dual one so that the conjugate gradient method can be used. The R-linear convergence rate is proved for an inexact globally convergent variant of the method. Numerical experiments conclude the paper. [ABSTRACT FROM AUTHOR]

This paper studied the range-constrained traffic assignment problem (RTAP), where heterogeneous range anxiety is considered among the driving population by electric vehicles (EVs). In order not to get stranded en-route, each EV driver is assumed to have his/her own driving range limit for being able to complete the trip. As a result, two types of multi-class RTAP can be defined through discrete or continuous distributed range anxiety. Given path-based side constraint structures, we proposed two variational inequality (VI) formulations for modeling discrete and continuous RTAPs, where the former is finite-dimensional according to a discrete number of user classes and the latter is infinite-dimensional accounting for an infinite number of user classes. We reformulate the continuous RTAP into finite-dimensional by merging adjacent EV drivers into one group. A unified path-based solution framework is developed to solve the two RTAPs, built upon the gradient projection algorithm. We design column generation and dropping schemes to adaptively maintain compact path sets and an inner equilibration strategy to accelerate convergence. Finally, a small network is used to examine the correctness and effectiveness of proposed models, and a large Winnipeg network is adopted to evaluate the impacts of stochastic driving range on network flows and computation costs. Numerical results provide compelling evidence of the outstanding superiority of the proposed algorithm, and show that EV drivers with heightened sensitivity towards range anxiety may contribute to the emergence of critical links experiencing severe congestion. • General VI formulations for range-constrained traffic assignment problem (RTAP). • Unified path-based solution framework is developed to solve two types of RTAPs. • We design column management and inner equilibration strategies. • Model features and solution efficiency of RTAPs are illustrated using two networks. [ABSTRACT FROM AUTHOR]

In this paper, optimal control strategies for temperature trajectory determination in order to maximize thermophilic bacteria in a fed-batch solid-state fermentation reactor are proposed. This process is modeled by nonlinear differential equations, which has been previously validated experimentally with scale reactor temperature profiles. The dynamic input aeration rate of the reactor is determined to increase microorganisms growth of a selective substrate for edible mushroom cultivation. In industrial practice, the process is comprised of three thermal stages with constant input air flow and three types of microorganisms in a 150-hour lapse. Scytalidium thermophilum and actinobacteria are desired in order to obtain a final biomass composition with acceptable microorganisms concentration. The Steepest Descent gradient algorithm in continuous time and the Gradient Projection algorithm in discrete-time are used for the process optimal control. A comparison of simulation results in the presence of disturbances is presented, where the resulting temperature trajectories exhibit similar tendencies as industrial data.

Gradient projection rotation (GPR) is an openly available and promising tool for factor and component rotation. We compare GPR toward the Varimax criterion in principal component analysis to the built-in Varimax procedure in SPSS. In a simulation study, we tested whether GPR-Varimax yielded multiple local solutions by creating population simple structure with a single optimum and with two optima, a global and a local one (double-optimum condition). The other conditions comprised the number of components (k = 3, 6, 9, and 12), the number of variables per component (m/k = 4, 6, and 8), the number of iterations per rotation (i = 25 and 250), and whether loadings were Kaiser normalized before rotation or not. GPR-Varimax was conducted with unrotated and multiple (q = 1, 10, 50, and 100) random start loadings. We found equal results for GPR-Varimax and SPSS-Varimax in most conditions. The few very small differences in favor of SPSS-Varimax were eliminated when Kaiser-normalized loadings and 250 iterations per rotation were used. Selecting the best solution out of multiple random starts in GPR-Varimax increased proximity to population components in the double-optimum condition with Kaiser normalized loadings, for which GPR-Varimax recovered population structure better than SPSS-Varimax. We also included an empirical example and found that GPR-Varimax and SPSS-Varimax yielded highly similar solutions for orthogonal simple structure in a real data set. We suggest that GPR-Varimax can be used as an alternative to Varimax rotation in SPSS. Users of GPR-Varimax should allow for at least 250 iterations, normalize loadings before rotation, and select the best solution from at least 10 random starts to ensure optimal results.

This paper presents three energy-efficient power allocation (PA) schemes for the downlink of a distributed multiple-input multiple-output (D-MIMO) system with beamforming at distributed antenna ports and antenna selection (AS) at a mobile station over composite Rayleigh channels. Subject to maximum power constraints, a constrained maximization problem on energy efficiency (EE) is formulated. Based on theoretical analysis, the optimal beamforming is shown to be aligned with the channel direction that renders the joint design problem to be a PA optimization one. It is shown that the optimization problem is strictly pseudo-concave with a unique maximum. Thus, an optimal PA scheme based on a gradient projection method is proposed to obtain the global solution. To speed up the gradient projection method, a closed-form initialization is derived. Furthermore, a low-complexity optimal PA schemes based on the Karush–Kuhn–Tucker conditions is proposed. This optimal scheme has a closed-form PA expression for a certain condition or else computes the PA by the Lambert function or the bisection method, and gives the same EE performance as that of the optimal gradient-based scheme. The performances of these two optimal schemes are optimal and verified to be valid in contrast to the inaccurate optimal scheme in the literature. For large maximum power, the closed-form initial PA will become the optimal solution and can be considered as a suboptimal solution with no iterative computation. Making use of the tight upper and lower bounds of the Lambert function, a very low-complexity suboptimal PA scheme is developed. Simulation results show that the proposed suboptimal scheme can achieve nearly optimal performance. Maximum EE and maximum channel power, which are optimal and suboptimal criteria, are adopted for antenna selection. The simple suboptimal criterion can provide EE performance very close to that of the optimal one. The D-MIMO with AS is shown to give considerable EE gain over the single receive antenna system. Moreover, the proposed three PA schemes can be used for multiuser diversity D-MIMO systems to achieve superior EE performance. [ABSTRACT FROM AUTHOR]

This study considers a generalized constant envelope-based precoding design problem in massive multiuser multiple-input multiple-output systems, in which the output signals of digital-to-analog converters are restricted to $M$ -ary phase-shift keying signaling to facilitate the use of low-cost and power-efficient power amplifiers at a base station. We aim to jointly optimize the precoding vector and the precoding factor to minimize the mean-squared error between the transmitted and the estimated symbols. However, the considered precoding problem is nonconvex and challenging to solve because the precoding vector and the precoding factor are coupled together. To address this problem, we apply the alternating minimization (AltMin) framework to decouple the considered problem into two problems: the precoding factor design problem and the precoding vector design problem. The former has an analytical solution, while the latter requires solving a difficult combinatorial problem, which we address by proposing a simple yet efficient gradient projection (GP)-based algorithm to find an approximate precoding vector. Simulation results demonstrate that the proposed GP-based AltMin algorithm achieves the same or higher BER performance than state-of-the-art algorithms while requiring lower complexity. In addition, when higher order modulations are employed, the error floor experienced by the existing algorithms can be substantially mitigated by the proposed GP-AltMin algorithm through increasing the number of phase bits. [ABSTRACT FROM AUTHOR]

Gradient projection rotation (GPR) is an openly available and promising tool for factor and component rotation. We compare GPR toward the Varimax criterion in principal component analysis to the built-in Varimax procedure in SPSS. In a simulation study, we tested whether GPR-Varimax yielded multiple local solutions by creating population simple structure with a single optimum and with two optima, a global and a local one (double-optimum condition). The other conditions comprised the number of components (k = 3, 6, 9, and 12), the number of variables per component (m/k = 4, 6, and 8), the number of iterations per rotation (i = 25 and 250), and whether loadings were Kaiser normalized before rotation or not. GPR-Varimax was conducted with unrotated and multiple (q = 1, 10, 50, and 100) random start loadings. We found equal results for GPR-Varimax and SPSS-Varimax in most conditions. The few very small differences in favor of SPSS-Varimax were eliminated when Kaiser-normalized loadings and 250 iterations per rotation were used. Selecting the best solution out of multiple random starts in GPR-Varimax increased proximity to population components in the double-optimum condition with Kaiser normalized loadings, for which GPR-Varimax recovered population structure better than SPSS-Varimax. We also included an empirical example and found that GPR-Varimax and SPSS-Varimax yielded highly similar solutions for orthogonal simple structure in a real data set. We suggest that GPR-Varimax can be used as an alternative to Varimax rotation in SPSS. Users of GPR-Varimax should allow for at least 250 iterations, normalize loadings before rotation, and select the best solution from at least 10 random starts to ensure optimal results. [ABSTRACT FROM AUTHOR]

This letter considers the use of alternating minimization (AltMin) framework as the main design principle in determining the optimal weights of hybrid digital and analog beamforming in millimeter-wave massive multiple-input multiple-output systems. However, such framework experiences analog beamformer design problem because analog beamformers are normally constructed using phase shifters (PSs), which result in non-convex constraints, and known solutions suffer from high computational complexity. We address this issue by proposing a simple yet efficient gradient projection (GP)-based AltMin (GP-AltMin) algorithm for hybrid beamforming design problems. In addition, when quantized PSs are adopted, we propose non-uniform PSs on the basis of the Lloyd–Max algorithm for minimizing the performance loss caused by quantization. Simulation results demonstrate that the proposed GP-AltMin algorithm not only has lower complexity but can also achieve spectral efficiency, which is comparable with or higher than those of the state-of-the-art algorithms that use unquantized and low-resolution PSs. [ABSTRACT FROM AUTHOR]

We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the gradient projection conjugate gradient (GPCG) algorithm for bound-constrained convex quadratic programming [J. J. Moré and G. Toraldo, SIAM J. Optim., 1 (1991), pp. 93{113], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained mini-mization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a com-parison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportional iterate, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in the case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

• A new DL-based supervised classification method is proposed that utilizes multiple undercomplete dictionaries. • An algorithm for learning undercomplete dictionaries from a set of labeled training data based on singular value decomposition is introduced. • Simulation results show that this method achieves higher classification accuracies compared with some of the similar methods, while having less computational complexity In dictionary-learning-based classification methods, a given data point is classified based on its representation over one or possibly more learned dictionaries. The goal is to find dictionaries that minimize the classification error. Previous works aimed to train dictionaries with representation and classification powers by using overcomplete dictionaries and sparse coding. These approaches are computationally expensive and do not scale readily to problems with high dimensional data. This paper presents a dictionary-learning-based classification method with the primary goal of classification and not representation. We propose to train multiple undercomplete dictionaries (one for each class of the problem). Each dictionary approximates the given test data, and the one with the lowest reconstruction error determines the class. Singular value decomposition (SVD) is used to obtain a straightforward algorithm for the resulted optimization problem. Simulation results show that our method achieves a higher accuracy compared with a number of successful sparse representation based classification methods, while having a significantly lower computational cost. [ABSTRACT FROM AUTHOR]