Series Of Subproblems Research Articles

Simulation-Based Design of a Cam-Driven Hydraulic Prosthetic Ankle

Background/Objectives: A cam-driven hydraulic prosthetic ankle was designed to overcome the weaknesses of commercial prostheses and research prototypes, which largely fail to mimic the energy-recycling behaviour of an intact ankle, resulting in poor walking performance for lower-limb prosthesis users. Methods: This novel device exploits miniature hydraulics to capture the negative work performed during stance, prior to push-off, in a hydraulic accumulator, and return positive work during push-off for forward body propulsion. Two cams are used to replicate intact ankle torque profiles based on experimental data. The design process for the new prosthesis used a design programme, implemented in MATLAB, based on a simulation of the main components of the prosthetic ankle. Results: In this paper, we present the design programme and explain how it is used to determine the cam profiles required to replicate intact ankle torque, as well as to size the cam follower return springs. Moreover, a constraint-based preliminary design investigation is described, which was conducted to size other key components affecting the device’s size, performance, and energy efficiency. Finally, the feasible design alternatives are compared in terms of their energy losses to determine the best design with regard to minimising both energy losses and device size. Conclusions: Such a design approach not only documents the design of a particular novel prosthetic ankle, but can also provide a systematic framework for decomposing complex design challenges into a series of sub-problems, providing a more effective alternative to heuristic approaches in prosthetic design.

Image reconstruction based on TV-L1 model for planar array capacitance defect detection

Planar array capacitive imaging is an emerging nondestructive testing (NDT) technology with broad application prospects. The severe ill-posed inverse problem of planar array capacitive imaging reduces quality of reconstructed images. In this paper, a total variation regularization model with an L1-norm (TV-L1 model) of 3 × 4 planar array capacitance imaging is established for imaging the defects detected in the inner layers of multi-layer composite components. Considering the solution instability caused by the non-differentiability of the TV-L1 model, a variable splitting self-adaptive augmented Lagrange alternating direction algorithm (VS-SALAD) is proposed. An augmented Lagrange method enhances model splitability, and a series of subproblems are obtained by variable splitting operation. The subproblems are solved iteratively using alternating minimization method with self-adaptive penalty parameter. Experimental results demonstrate that TV-L1 model effectively alleviates the ill-posed problem, and the proposed algorithm significantly improves both the quality of reconstructed images and solving speed of the TV-L1 model.

Binary pattern retrieval with Kuramoto-type oscillators via a least orthogonal lift of three patterns

Abstract Given a set of standard binary patterns and a defective pattern, the pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The Hebbian network of Kuramoto oscillators with second-order coupling provides a dynamical model for this task, and the mutual orthogonality in memorised patterns enables us to distinguish these memorised patterns from most others in terms of stability. For the sake of error-free retrieval for general problems lacking orthogonality, a unified approach was proposed which transforms the problem into a series of subproblems with orthogonality using the orthogonal lift for two patterns. In this work, we propose the least orthogonal lift for three patterns, which evidently reduces the time of solving subproblems and even the dimensions of subproblems. Furthermore, we provide an estimate for the critical strength for stability/instability of binary patterns, which is convenient in practical use. Simulation results are presented to illustrate the effectiveness of the proposed approach.

Terminal Trajectory Planning for Synthetic Aperture Radar Imaging Guidance Based on Chronological Iterative Search Framework.

Synthetic aperture radar (SAR) is capable of obtaining the high-resolution 2-D image of the interested target scene, which enables advanced remote sensing and military applications, such as missile terminal guidance. In this article, the terminal trajectory planning for SAR imaging guidance is first investigated. It is found that the guidance performance of an attack platform is determined by the adopted terminal trajectory. Therefore, the aim of the terminal trajectory planning is to generate a set of feasible flight paths to guide the attack platform toward the target and meanwhile obtain the optimized SAR imaging performance for enhanced guidance precision. The trajectory planning is then modeled as a constrained multiobjective optimization problem given a high-dimensional search space, where the trajectory control and SAR imaging performance are comprehensively considered. By utilizing the temporal-order-dependent property of the trajectory planning problem, a chronological iterative search framework (CISF) is proposed. The problem is decomposed into a series of subproblems, where the search space, objective functions, and constraints are reformulated in chronological order. The difficulty of solving the trajectory planning problem is thus significantly alleviated. Then, the search strategy of CISF is devised to solve the subproblems successively. The optimization results of the preceding subproblem can be utilized as the initial input of the subsequent subproblems to enhance the convergence and search performance. Finally, a trajectory planning method is put forward based on CISF. Experimental studies demonstrate the effectiveness and superiority of the proposed CISF compared with the state-of-the-art multiobjective evolutionary methods. The proposed trajectory planning method can generate a set of feasible terminal trajectories with optimized mission performance.

Recurrent neural networks that learn multi-step visual routines with reinforcement learning.

Many cognitive problems can be decomposed into series of subproblems that are solved sequentially by the brain. When subproblems are solved, relevant intermediate results need to be stored by neurons and propagated to the next subproblem, until the overarching goal has been completed. We will here consider visual tasks, which can be decomposed into sequences of elemental visual operations. Experimental evidence suggests that intermediate results of the elemental operations are stored in working memory as an enhancement of neural activity in the visual cortex. The focus of enhanced activity is then available for subsequent operations to act upon. The main question at stake is how the elemental operations and their sequencing can emerge in neural networks that are trained with only rewards, in a reinforcement learning setting. We here propose a new recurrent neural network architecture that can learn composite visual tasks that require the application of successive elemental operations. Specifically, we selected three tasks for which electrophysiological recordings of monkeys' visual cortex are available. To train the networks, we used RELEARNN, a biologically plausible four-factor Hebbian learning rule, which is local both in time and space. We report that networks learn elemental operations, such as contour grouping and visual search, and execute sequences of operations, solely based on the characteristics of the visual stimuli and the reward structure of a task. After training was completed, the activity of the units of the neural network elicited by behaviorally relevant image items was stronger than that elicited by irrelevant ones, just as has been observed in the visual cortex of monkeys solving the same tasks. Relevant information that needed to be exchanged between subroutines was maintained as a focus of enhanced activity and passed on to the subsequent subroutines. Our results demonstrate how a biologically plausible learning rule can train a recurrent neural network on multistep visual tasks.

Dynamic reconfigurable intelligent surface deployment for physical layer security enhancement in mmWave systems

AbstractIn next‐generation wireless communications, reconfigurable intelligent surface (RIS) has emerged as a cost‐effective technique for enhancing physical layer security (PLS) in millimeter‐wave (mmWave) communications, especially under challenging scenarios with adversarial entities and obstructions. However, the primary studies for RIS incorporation in mmWave communication systems utilized static deployments, lacking adaptability and efficiency in complex environments. To address this problem, a dynamic RIS deployment design framework is introduced for PLS enhancement in mmWave systems against jamming and eavesdropping attacks. For the design, it is aimed to maximize the secrecy rate by jointly optimizing the RIS selection with the beamforming design. The resulting optimization problem is challenging to solve due to the coupling of the RIS control factor, joint beamforming design, and non‐convex constraints. To tackle these issues, an efficient multi‐RIS‐aided PLS enhancement algorithm is proposed. It transforms the objective into a series of subproblems and employs the fractional programming technique and prox‐linear block coordinate descent updating method to solve them alternatively and obtain the optimal solution. The simulations demonstrate the advantage of the dynamic deployment, which exhibits enhanced security performance with reduced complexity compared with benchmarks. Further examinations also provide insight into optimal RIS activation configurations, achieving optimal balance for securing mmWave communications against emerging threats while maintaining system efficiency.

New Exact Algorithm for the integrated train timetabling and rolling stock circulation planning problem with stochastic demand

This paper studies an integrated train timetabling and rolling stock circulation planning problem with stochastic demand and flexible train composition (TRSF). A novel stochastic integer programming model, which is formulated on a space-time underlying network to simultaneously optimize the train timetable and rolling stock circulation plan with flexible train composition, is proposed by explicitly considering the random feature of passenger distribution on an urban rail transit line. To solve this problem efficiently, the proposed model is decomposed into a master problem and a series of sub-problems regarding different stochastic scenarios. We further prove that each sub-problem model is equivalent to its linear programming relaxation problem, by proving that the coefficient matrix of each linear programming relaxation model is totally unimodular. Then, the classical Benders decomposition algorithm is applied to the studied problem. Based on the model characteristics, both single-cut and multi-cut methods with some speed-up techniques are developed to solve the proposed model in a novel and effective way. Numerical experiments are conducted on small-scale cases and large-scale cases derived from Shanghai Metro Line 17, and the results show that solving the stochastic problem can extract gains in efficiency and the value of stochastic solution tends to be high.

Communication and computational resource optimization for Industry 5.0 smart devices empowered by MEC

Smart devices in Industry 5.0, such as sensors and robots, are often limited by low battery life and finite computational resources, hindering their ability to perform complex tasks. By offloading computation-intensive tasks to Mobile Edge Cloud Computing (MEC) servers at the network’s edge, businesses can achieve real-time data processing and analysis, reducing communication latency, quicker response times, and improved system reliability. This work presents an integrated framework for MEC and Industry 5.0, aimed at enhancing the performance, efficiency, and flexibility of industrial processes. In particular, we propose a joint optimization problem that maximizes computational energy efficiency by optimally allocating resources, such as processing power and computational resources, as well as device association, in the most efficient manner possible. The problem is formulated as nonconvex/nonlinear, which is intractable and poses high complexity. To solve this challenging problem, we first transform and decouple the original optimization problem into a series of subproblems using the block coordinate descent method. Then, we iteratively obtain an efficient solution using convex optimization methods. In addition, our work sheds light on the fundamental trade-off between local computation and partial offloading schemes. The results show that for small data size requirements, the performance is comparable among different schemes. However, as data size increases, our proposed hybrid scheme, which includes a partial offloading scheme, outperforms others, highlighting the effectiveness of the proposed joint optimization scheme.

Optimal instant discounts of multiple ride options at a ride-hailing aggregator

Recently, ride-hailing aggregators have emerged to help passengers find the best ride options. It aggregates the results from different service providers, allowing sorting and filtering all in a single app. It is not unusual for an aggregator to offer instant discounts on the prices returned from the service providers to increase its Gross Transaction Value (GTV) or Total Transaction Volume (TTV). In this research, we study this optimal instant discount problem, which can be viewed as a new variant of the classical multi-product price optimization problem. We first use the Nested Logit model to predict the probability a passenger would complete the trip with each ride option. We then formulate the instant discount problem as a nonlinear optimization model with a budget constraint and a group of discount bound constraints. To solve this model, we construct a surrogate relaxation formulation with strong duality. We develop a Lagrangian-dual-based approach to decompose this problem into a series of subproblems, and then design heuristic methods to give feasible solutions. For both cases when the GTV or the TTV is maximized, we quantify the optimality gap, give its asymptotic properties, and establish conditions under which it becomes zero and tight. Finally, we use real data from Meituan, a leading ride-hailing aggregator in China, to validate the proposed approach. Results show that compared to the baseline methods, we can improve the GTV by 1.293%, improve the TTV by 0.475%, and decrease the magnitude of the optimality gap.

A column generation approach for the team formation problem

We address a cooperative multi-team formation problem where there are multiple teams in an organization, and teams can be reformed by exchanging members among each other. We formulate this problem as a nonlinear integer programming model, by later reformulating it, which prescribes the optimal exchange decisions for all teams that maximizes the minimum resulting team value across all teams. For this problem, in order to obtain tighter dual bounds, we propose a column generation approach where each column represents specific exchange decisions of a team pair, and at each iteration, a set of attractive exchange patterns for all team pairs are generated by a series of subproblems. We implement a parallel processing scheme to solve all subproblems in order to further accelerate the procedure. In addition, to obtain near-optimal solutions, we propose a column generation-based methodology, where at the last iteration, the restricted master problem is solved as an integer programming model with all generated columns. Our results show that the column generation procedure provides dual bound improvements ranging between 2.44% and 5.31%, on average, over the linear programming relaxation of the original model for our generated instances. In addition, the proposed column generation-based methodology yields near-optimal, and in most cases, optimal solutions by providing CPU savings ranging from 71.6% to 89.2%, on average, over the CPU times of the original integer model.

Tensor-Based Joint Beamforming with Ultrasonic and RIS-Assisted Dual-Hop Hybrid FSO mmWave Massive MIMO of V2X

Reconfigurable intelligent surface (RIS)-assisted millimeter-wave (mmWave) communication systems relying on hybrid beamforming structures are capable of achieving high spectral efficiency at a low hardware complexity and with low power consumption. Tensor-based joint beamforming with low-cost ultrasonic and RIS-assisted Dual-Hop Hybrid free space optical (FSO) mm Wave massive Multiple Input Multiple Output (MIMO) of vehicle-to-everything (V2X) is proposed. To address the occlusion problem for high-speed mobility of the vehicle, an RIS-assisted mixed FSO-MIMO V2X system is proposed. The low-cost ultrasonic array signal model is developed to solve the accurate direction-of-arrival (DOA) estimation. The ultrasonic-assisted RIS phase shift matrix based on subspace self-organizing iterations is designed to track the beam direction between RIS and vehicle. Specifically, the associated bandwidth-efficiency maximization problem is transformed into a series of subproblems, where the subarray of phase shifters and RIS elements is jointly optimized to maximize each subarray’s rate. The vehicle motion state is transformed into a two-dimensional model for prior distribution to calculate the particle weights of the RIS phase. Multi-vehicle Tucker tensor decomposition is used to describe the high-dimensional beam space. We conceive a multi-vehicle joint optimization method for designing the hybrid beamforming matrix of the base station (BS) and the passive beamforming matrix of the RIS. A cascaded channel decomposition method based on Singular Value Decomposition (SVD) is used to obtain the combined matrix beamforming of BS and vehicle. Our simulation results demonstrate the superiority of the proposed method compared to its traditional counterparts.

Quadratically Constrained Linear Programming-based energy-efficient driving for High-speed Trains with neutral zone and time window

Among the numerous challenges that are faced during the daily operation of high-speed rail, the management of the neutral zones in the catenary during unexpected disturbances remains underinvestigated. This study proposes a computerised method for managing the operations of a high-speed train with regenerative braking for passing neutral zones under disturbance to minimise energy consumption. For the first time, different mechanism-based Automatic Passing Neutral Zone systems, including the Magnetic Induction System and Automatic Train Protection, are analysed and modelled as location-based and time-based constraints, respectively. Motion constraints caused by disturbances are described by time windows. Forced coasting and air brake-allowed passing neutral zone rules are considered in these models. The original nonlinear model with location-based constraints is transcribed as a quadratically constrained linear model and then solved. The optimality consistency and its establishment condition between the original and transcribed models are analysed based on the Karush–Kuhn–Tucker conditions. A high-quality solution is obtained when the establishment condition holds. The model with time-based constraints is novelly transformed into an optimal switching point problem. A series of sub-problems are iteratively and efficiently solved. Comprehensive experiments are conducted based on practical data from a high-speed rail system in China. The proposed method is significantly beneficial when compared to Mixed-integer Linear Programming and Artificial Driving Algorithms. Moreover, the impacts of different mechanism-based automatic passing neutral zone systems, operation rules, and a combination of time window settings are extensively analysed.

Simultaneously Transmitting and Reflecting Surface (STARS) for Terahertz Communications

A simultaneously transmitting and reflecting surface (STARS) aided terahertz (THz) communication system is proposed. A novel power consumption model is proposed that depends on the type and resolution of the STARS elements. The spectral efficiency (SE) and energy efficiency (EE) are maximized in both narrowband and wideband THz systems by jointly optimizing the hybrid beamforming at the base station (BS) and the passive beamforming at the STARS. 1) For narrowband systems, independent phase-shift STARSs are investigated first. The resulting complex joint optimization problem is decoupled into a series of subproblems using penalty dual decomposition. Low-complexity element- wise algorithms are proposed to optimize the analog beamforming at the BS and the passive beamforming at the STARS. The proposed algorithm is then extended to the case of coupled phase-shift STARS. 2) For wideband systems, the spatial wideband effect at the BS and STARS leads to significant performance degradation due to the beam split issue. To address this, true time delayers (TTDs) are introduced into the conventional hybrid beamforming structure for facilitating wideband beamforming. An iterative algorithm based on the quasi-Newton method is proposed to design the coefficients of the TTDs. Finally, our numerical results confirm the superiority of the STARS over the conventional reconfigurable intelligent surface (RIS). It is also revealed that i) there is only a slight performance loss in terms of SE and EE caused by coupled phase shifts of the STARS in both narrowband and wideband systems, and ii) the conventional hybrid beamforming achieves comparable SE performance and much higher EE performance compared with the full-digital beamforming in narrowband systems but not in wideband systems, where the TTD-based hybrid beamforming is required for mitigating wideband beam split.

Fuzzy style flat-based clustering

The recently developed fuzzy style k-plane clustering (S-KPC) algorithm displays promising clustering quality by leveraging both similarities and distinguishable styles between samples on stylistic data. However, S-KPC becomes vulnerable to similar styles that are not easily distinguishable. In this study, a novel fuzzy style flat-based clustering (FSFC) algorithm is proposed to overcome this vulnerability. In FSFC, a style flat matrix (SFM) is designed to project samples onto appropriate flats while maintaining the styles of different clusters in a reasonable manner. Based on SFM, the core of FSFC is to learn the potentially intersecting manifold structures of clusters in the projected flat space to make samples with the same style close to the cluster center and simultaneously far away from the other cluster centers. Furthermore, the objective function of FSFC can provide scale flexibility for each flat in the projected flat space. In particular, the optimization problem of FSFC can be decomposed into a series of sub-problems about the flat parameters, which can be locally optimized using the concave-convex procedure (CCCP). Extensive experiments on both synthetic and real-world datasets demonstrate the competitive clustering performance of FSFC. Moreover, FSFC outperforms some state-of-the-art manifold clustering algorithms on six case studies about stylistic data.

Robust Hybrid Beamforming Design for Multi-RIS Assisted MIMO System With Imperfect CSI

Reconfigurable intelligent surface (RIS) has been developed as a promising approach to enhance the performance of fifth-generation (5G) systems through intelligently reconfiguring the reflection elements. However, RIS-assisted beamforming design highly depends on the channel state information (CSI) and RIS’s location, which could have a significant impact on system performance. In this paper, the robust beamforming design is investigated for a RIS-assisted multiuser millimeter wave system with imperfect CSI, where the weighted sum-rate maximization problem (WSM) is formulated to jointly optimize transmit beamforming of the BS, RIS placement and reflect beamforming of the RIS. The considered WSM maximization problem includes CSI error, phase shifts matrices, transmit beamforming as well as RIS placement variables, which results in a complicated nonconvex problem. To handle this problem, the original problem is divided into a series of subproblems, where the location of RIS, transmit/reflect beamforming and CSI error are optimized iteratively. Then, a multiobjective evolutionary algorithm is introduced to gradient projection-based alternating optimization, which can alleviate the performance loss caused by the effect of imperfect CSI. Simulation results reveal that the proposed scheme can potentially enhance the performance of existing wireless communication, especially considering a desirable trade-off among beamforming gain, user priority and error factor.

Resource Allocation for Cell-Free Massive MIMO-Enabled URLLC Downlink Systems

Ultra-Reliable and low-latency communication (URLLC) is a pivotal technique for enabling the wireless control over industrial Internet-of-Things (IIoT) devices. By deploying distributed access points (APs), cell-free massive multiple-input and multiple-output (CF mMIMO) has great potential to provide URLLC services for IIoT devices. In this paper, we investigate CF mMIMO-enabled URLLC in a smart factory. Lower bounds (LBs) of downlink ergodic data rate under finite channel blocklength (FCBL) with imperfect channel state information (CSI) are derived for maximum-ratio transmission (MRT), full-pilot zero-forcing (FZF), and local zero-forcing (LZF) precoding schemes. Meanwhile, the weighted sum rate is maximized by jointly optimizing the pilot power and transmission power based on the derived LBs. Specifically, we first provide the globally optimal solution of the pilot power, and then introduce some approximations to transform the original problems into a series of subproblems, which can be expressed in a geometric programming (GP) form that can be readily solved. Finally, an iterative algorithm is proposed to optimize the power allocation based on various precoding schemes. Simulation results demonstrate that the proposed algorithm is superior to the existing algorithms, and that the quality of URLLC services will benefit by deploying more APs, except for the FZF precoding scheme.

A Warm-Start Strategy in Interior Point Methods for Shrinking Horizon Model Predictive Control With Variable Discretization Step

For optimal control problems with random external disturbances, offline algorithms may not perform well because the state cannot be predicted. To solve the problems online, the model predictive control is frequently used, where a series of subproblems with different initial conditions are solved. These convex subproblems could be solved by using interior point methods after discrete approximations. To obtain a solution as accurately as possible under limited computing resources, the variable discretization step is usually considered. To improve efficiency, warm-start methods are considered, which utilize the similarity of subproblem structures. Specifically, the information obtained by solving earlier subproblems is used to solve subsequent subproblems. However, the discretization steps and the sampling numbers of subproblems may be different, and the initial states vary greatly, resulting in poor performance of existing warm-start strategies for interior point methods. In this paper, we present a new warm-start strategy for interior-point methods. In the strategy, a convex combination of components of the earlier optimal solution is used to construct an interpolation point, and we use the convex combination of the modified interpolation point and the cold-start point to obtain a warm-start point. We prove that the worst-case iteration complexity of our strategy is better than that of the cold-start. In the numerical experiment of fuel-optimal planetary powered-descent guidance problems, our strategy reduces the number of iterations of second-order cone programming by about <inline-formula><tex-math notation="LaTeX">$80\%$</tex-math></inline-formula> with the number of samples available in practical applications.

Hybrid precoding design for secure smart-grid enabled MIMO wireless communications in Industry 5.0

Secure and reliable wireless communication systems are essential for the successful integration of smart grid-enabled wireless communications and Industry 5.0 applications. This work proposes a hybrid precoding design for secure smart-grid multiple-input multiple-output (MIMO) wireless communications in Industry 5.0 to maximize the average secrecy capacity of the system while minimizing user outages crucial for maintaining the security and privacy of sensitive data in critical infrastructure. To achieve this, we develop a mathematical model that considers the impact of various system parameters, such as eavesdropping and interference caused by the active eavesdropper and noise, on the secrecy capacity. To find an efficient solution to the optimization problem, we propose an algorithm that decouples the original optimization problem into a series of subproblems and uses iterative techniques to find the optimal values, thereby improving computational efficiency. The hybrid precoding scheme is an effective technique for optimizing the design parameters of MIMO-based secure wireless communication systems. Our proposed approach provides a practical solution for achieving this optimization. Our numerical results demonstrate that our proposed scheme outperforms traditional benchmark schemes, maximizing the average secrecy capacity while minimizing user outages. Our work highlights the importance of secure wireless communication systems in Industry 5.0 and smart grid applications. The proposed approach provides an efficient method for designing secure wireless communication systems that can effectively address the unique challenges posed by these critical infrastructure systems.

Computational energy efficient trajectory planning for UAV-enabled 6G MEC communication network

In this work, we consider the unmanned aerial vehicle (UAV) as a flying MEC server to provide computational and communication resources to low-power sensor nodes. These sensor nodes are deployed uniformly over the predefined area. Due to conjunction or natural disaster, the central cloud system is not available. As a result, the quality of service is highly compromised. To ensure communication, UAV-MEC emerges as an effective solution to provide on-demand services. Therefore, by considering the limited serving capacity of UAVs, in this work, we formulate a joint computational energy efficiency problem by jointly optimizing the trajectory, computation, and communication resources. Furthermore, to address the computational complexity, we decouple the original optimization problem into a series of sub-problems and iteratively solve them. Moreover, extensive simulations were carried out to validate the proposed solution, and average results were produced. The numerical results demonstrate the proposed scheme’s effectiveness compared to other benchmark schemes.

Adaptive sparsity-regularized deep dictionary learning based on lifted proximal operator machine

Deep dictionary learning (DDL) can mine deeper representations of data more effectively than single-layer dictionary learning. However, existing DDL methods with specific sparse regularizers lead to designated deep sparse representations. Existing DDL optimization methods have limited feature extraction capability because their nonlinear functions must satisfy the condition that the functions are invertible. This paper presents a new DDL model with a learned sparsity constraint and a noninvertible soft-thresholding (ST) function, where the learned sparsity constraint can obtain a data-driven sparse representation and the ST (that produces a sparse output as a nonlinear function) can enhance its the feature extraction capability. To solve the optimization problem efficiently with the noninvertible and learned sparsity constraints, we propose to employ a lifted proximal operator machine to transform the proposed DDL problem into a series of subproblems that include sparsity-regularized and convex minimizations. For the sparsity-regularized minimization, we derive the parameterized proximal operator of the sparse regularizer, which is considered as the activation function used to construct the network. The parameters of the activation function are trained by backpropagation, thus obtaining the proximal operator of the learnable sparse regularizer simultaneously with a sparse solution. The convex minimization for the dictionaries and the coefficients can be obtained via the accelerated proximal gradient and the optimal condition, respectively. In the numerical classification and reconstruction experiments, the proposed algorithm outperformed existing DDL algorithms in terms of classification accuracy, image reconstruction, and noise immunity.