Iterative Algorithm Research Articles

Incoherent frequency-selective multiplexing holographic encryption by using a triple-beam setup

Inspired by the wavefront masking of the scattering medium, we proposed a multiplexed coded aperture holographic encryption method. The incoherent multiplexed phase mask encryption experiments involved in the method are realized for what we believe to be the first time. From the holograms, we extracted three images using the frequency-selective phase iterative coding algorithm we purposely put forward. Through the theoretical analysis of frequency-selective phase coding, the constructed linear system can recover the original image well. We verified the security of the cryptography using structural similarity index measurements, key space, key sensitivity, and axial multiplexing. Through the theoretical analysis of frequency-selective phase coding, we think the present triple-multiplexed cryptography can be directly extended to multiple-multiplexed cases and applied to multi-image encryption. It may cause a paradigm shift in applications such as incoherent image encryption, wavefront acquisition in scattering, information hiding, and fast target identification.

A novel treatment planning method via scissor beams for uniform-target-dose proton GRID with peak-valley-dose-ratio optimization.

Proton spatially fractionated RT (SFRT) can potentially synergize the unique advantages of using proton Bragg peak and SFRT peak-valley dose ratio (PVDR) to reduce the radiation-induced damage for normal tissues. Uniform-target-dose (UTD) proton GRID is a proton SFRT modality that can be clinically desirable and conveniently adopted since its UTD resembles target dose distribution in conventional proton RT (CONV). However, UTD proton GRID is not used clinically, which is likely due to the lack of an effective treatment planning method. This work will develop a novel treatment planning method using scissor beams (SB) for UTD proton GRID, with the joint optimization of PVDR and dose objectives. The SB method for spatial dose modulation in normal tissues with UTD has two steps: (1) a primary beam (PB) is halved with interleaved beamlets, to generate spatial dose modulation in normal tissues; (2) a complementary beam (CB) is added to fill in previously valley-dose positions in the target to generate UTD, while the CB is angled slightly from the PB, to maintain spatial dose modulation in normal tissues. A treatment planning method with PVDR optimization via the joint total variation and L1 (TVL1) regularization is developed to jointly optimize PVDR and dose objectives. The plan optimization solution is obtained using an iterative convex relaxation algorithm. The new methods SB and SB-TVL1 were validated in comparison with CONV. Compared to CONV of relatively homogeneous dose distribution, SB had modulated spatial dose pattern in normal tissues with UTD and comparable plan quality. Compared to SB, SB-TVL1 further maximized PVDR, with comparable dose-volume parameters. A novel SB method is proposed that can generate modulated spatial dose pattern in normal tissues to achieve UTD proton GRID. A treatment planning method with PVDR optimization capability via TVL1 regularization is developed that can jointly optimize PVDR and dose objectives for proton GRID.

Stochastic modeling and optimization of discrete-time cold standby repairable systems with unreliable repair facility and retrial mechanism

This study develops novel reliability models for discrete-time cold standby repairable systems with unreliable repair facility and retrial mechanism. In view of the situation that the state of some engineering systems cannot be continuously monitored, discrete distribution is used to describe the distribution of random variables involved in this paper. For the case that multiple events can occur simultaneously in the discrete-time system model, the priority order of multiple events occurring simultaneously is defined. Two different models are proposed based on two types of priority rules. The retrial mechanism of the components is considered in the case that the failure information of the system components cannot be successfully transmitted to the repair facility. The preventive maintenance (PM) and repair after the breakdown of the repair facility are considered. The key reliability indices are derived by using the iterative algorithm of the difference equation and the generative function method. The algorithms for calculating system state probability, key reliability indices and cost–benefit ratio (CBR) are designed. To demonstrate the impact of each parameter on the system reliability indices, numerical results are provided. The optimal repair rate configuration scheme is chosen using the Particle Swarm Optimization (PSO) algorithm to achieve the lowest possible CBR. The impact of the system model with or without PM on stationary availability and CBR is shown. In addition, the impact of the number of system components on a series of system performance indices is analyzed, and the optimal number of components of the system corresponding to the minimum value of CBR is obtained.

Energy transfer, conversion and mechanical regulation in graded piezoelectric semiconductor bipolar junction transistor

While the effective regulation of carrier redistribution in piezoelectric semiconductor devices by mechanical loads through piezoelectric polarized electric fields is well-studied, there is a scarcity of reported research on the influence of energy transfer and conversion characteristics in a non-thermal equilibrium state of these devices. This paper establishes a nonlinear multi-field coupling model for piezoelectric semiconductor graded bipolar junction transistor (PS-GBJT) and proposes a corresponding multi-gradient iterative algorithm. We regard the product of current and built-in electric field as the work done by electric field force on carriers. By incorporating the energy absorbed/released during carrier recombination/generation process, we observe that non-equilibrium carriers act as a medium that facilitates energy conversion between these two mechanisms. Additionally, the non-uniformly distributed doping extends the influence of space charge region throughout entire transistor. In terms of external input electrical energy, the emitter, base, and collector regions play roles of input energy, transfer energy, and output energy, respectively. The study concludes by demonstrating the regulation of energy transfer or conversion efficiency in emitter, base, and collector regions through mechanical loads, thereby altering the heating behavior of entire GBJT and reshaping its energy transfer path. This leads to the significant finding that distinct energy transfer paths fundamentally define the various operational states of GBJT. The novel insights gained from this research hold reference value for thermal design and energy management of electronic devices.

Coordinative dispatching of shared and public transportation under passenger flow outburst

In the passenger flow outburst area (e.g., metro emergency, dispersal of sporting events, and concerts), passengers put forward large demand for ride-sourcing services. Insufficient ride-sourcing supply of the area induces increasing matching time, and the ride-sourcing influx from outside picking up passengers leads to heavier road traffic congestion in the flow outburst area. To solve this problem, we propose a novel coordinative dispatching method of ride-sourcing and multi-modal transit, in which subsidies and buses are provided for passengers in flow outburst areas to encourage taking other public transportation to low passenger flow areas first and then taking ride-sourcing to their destinations. In this paper, there are two options for passengers: (a) wait for ride-sourcing to pick them up; (b) take other public transportation first and then take ride-sourcing. To obtain the optimal dispatching and subsidy schemes, we develop a bi-level mixed integer programming model based on network flow theory and design the corresponding iterative algorithm to solve it. Considering the high uncertainty of ride-sourcing demand in the area, we further develop a robust optimization model to obtain more reliable schemes. Case studies based on a real-problem-scale dataset are conducted. The results demonstrate that our approach can be carried out in real time, and encouraging passengers to take multi-modal transit shows great potential in reducing affected users’ delays. The robust optimization model offers a more reliable and competitive solution when demand varies widely. Our method offers a win–win–win way for passengers, ride-sourcing service providers, and public transportation systems.

One step inversion free iterative algorithm for computing generalised bisymmetric solution for nonlinear matrix equations

In this paper, we investigate one step inversion free iterative algorithm for computing generalised bisymmetric solution to the nonlinear matrix equations (NLME) X + A T X − q A + B T X − q B = H ( q ≥ 2 ) , where A , B , H are given matrices, while X is a generalised bisymmetric matrix to be computed. The necessary conditions for the existence of a generalised bisymmetric solution and the convergence of the suggested iterative method are derived. Some lemmas and theorems are provided and proved where the iterative solutions are obtained. Four numerical examples are presented to exhibit the effectiveness of the suggested method and to verify the theoretical findings of this paper.

Quantitative detection and sorting of broken kernels and chalky grains in milled rice using computer vision algorithms

The milled rice industry faces challenges in effectively detecting and sorting broken kernels and chalky grains. This paper presents a computer vision approach to detect and characterize broken kernels and chalky grains in three milled rice varieties (HM105, WYD4, and JH6). Broken kernel sorting was validated against an ISO-derived recognition method, while the identification of chalky grains was validated using a DSC to assess thermal behavior. The logistic regression showed a strong discriminatory ability in identifying broken kernels. Unlike the ISO method, logistic regression achieved high sorting accuracy for HM105, WYD4, and JH6. The iterative threshold segmentation algorithm also accurately segmented chalky regions in milled rice grains. Regression analysis revealed a significant positive correlation (R2 ≥ 0.94) between the chalky region area and ΔH values. Therefore, this study introduces a robust and rapid methodology for quantitatively detecting milled rice, with the potential for automation and intelligentization in rice-based food processes.

Trustful double auction design for Peer-to-Peer energy trading between interconnected micro-grids with supply–demand imbalance

Peer-to-Peer (P2P) energy trading between Interconnected Micro-Grids (IMGs) presents a promising approach for enhancing the economic advantages for prosumers while mitigating supply–demand imbalances within individual Micro-Grids (MGs). This paper proposes a trustful double auction mechanism for P2P energy trading among prosumers in IMGs, structured into two distinct stages. Initially, concurrent auctions are conducted within each MG to facilitate intra-grid trading, followed by subsequent auctions facilitating inter-grid trading across IMGs. We figure out the allocation and pricing rules that satisfy the required properties in mechanism design even when the auction consists of two stages. Given that energy transactions across IMGs entail non-negligible power losses, equitable allocation of these losses among prosumers is paramount. To address this, we integrate a fair cost distribution methodology into our auction mechanism, implemented by an iterative algorithm. Rigorous analysis substantiates our proposed auction mechanism’s incentive compatibility, individual rationality, and budget balance, thereby fostering truthful and voluntary prosumer participation while averting potential market deficits. Numerical analysis underscores the efficacy of our approach, showcasing a significant improvement in P2P transactions and supply–demand balance enhancement compared to trading solely within individual MGs.

A Gramian matrix approach to synthesizing finite-frequency H2 controller

This paper studies the H2 control problem for linear continuous-time systems over a finite-frequency range. Using the finite-frequency Gramian matrix approach, a necessary and sufficient condition is obtained for the characterization of the finite-frequency H2 performance of a Hurwitz stability system. With such a characterization, a sufficient condition for the solvability of the finite-frequency H2 control problem is derived. An iterative algorithm is then constructed to solve the corresponding controller gain numerically. Finally, a numerical example of a two-cart-one-spring system is given to illustrate the effectiveness of the proposed scheme.

Dynamic Flow Algorithm with Intermediate Storage for Emergency Evacuation Allowing Lane Reversal based on Incomplete Intuitionistic Fuzzy Preference Relation

Background Evacuation in case of disasters is of the greatest importance because of significant occurrences of natural and artificial disasters worldwide, which is why a reliable evacuation plan is always needed. However, evacuation models are difficult to develop due to various uncertain aspects that need to be considered, multiple and often conflicting criteria for evaluation and as lack of expertise regarding a specific preference of alternatives. Objective This study aims to transport the maximum number of evacuees in a dynamic network with lane reversal by a safe pattern of transportation, i.e., allowing storage at intermediate nodes. The optimal order of shelters and intermediate nodes for a reliable evacuation plan will be defined by incomplete intuitionistic fuzzy preference relation. Methods The illustrated method incorporates multiple and often conflicting criteria into a process of emergency decision-making. When evaluating evacuation alternatives, a decision-maker may hesitate and be unsure which alternative is better or not have sufficient expertise to evaluate a pair of alternatives. To model uncertainty and hesitation, intuitionistic fuzzy values are used to describe alternatives in more detail. This study relies on flow models and graph theory to simulate the movement of evacuees to safe destinations. Furthermore, fuzzy methods and their recent modifications are applied to determine the effective priority order of shelters. A case study which simulates the evacuation of aggrieved to safe destinations is presented. Results A method of evaluating the shelters and intermediate nodes for evacuation based on incomplete intuitionistic fuzzy preference relation is proposed. The method allows the missed values of experts’ assessments to be filled in regarding the evacuation alternatives and deals with intuitionistic fuzzy values, which describe experts’ hesitation. The dynamic character of flow distribution enables transit arc capacities and time factors to be processed. The contraflow technique, which is a powerful tool to decrease traffic jams and congestion on roads by reversing the movement along the unused segments, is applied to maximize the number of survivors. Conclusion The results of the method were compared to those of existing methods, and their consistency was proved. In the future, we intend to apply interval-valued intuitionistic preference relations and iterative algorithms to improve the consistency of intuitionistic preference relations to the tasks of transporting the maximum possible number of aggrieved to safe locations.

Fluid–Structure Interaction Analysis of Manta-Bots with Self-Induced Vertical Undulations during Fin-Based Locomotion

Driven by the demands of ocean exploration, an increasing number of manta ray-inspired robots have been designed and manufactured, primarily utilizing flexible skeletons combined with motor-driven mechanisms. However, the mechanical analysis of these designs remains underdeveloped, often relying on simplistic imitation of biological prototypes and typically neglecting the vertical motion induced by pectoral fin flapping. This paper presents a fluid–structure interaction analysis framework that couples rigid body motion with elastic deformation using flexible multibody dynamics and the vortex particle method. An implicit iterative algorithm with Aitken relaxation is employed to address added-mass instability, and the framework has been validated against experimental data. An analysis of a representative manta-bot model shows that self-induced vertical undulations reduce the thrust coefficient by approximately 40% compared to fixed vertical degrees of freedom, while slightly improving overall propulsive efficiency. The study also highlights the critical role of mass distribution in manta-bots, noting that excessive focus on complex pectoral fin movements and large fin mass can significantly reduce thrust by increasing vertical displacement, ultimately proving counterproductive.

Joint mining of fluid knowledge and multi-sensor data for gas–water two-phase flow status monitoring and evolution analysis

Gas-water two-phase flow process widely exists in petroleum, chemical and other industrial processes, which is intricate, manifested by dynamic evolution of fluid structures and real-time fluctuation of flow parameters. Detailed description and real-time monitoring of flow status are of great significance to safe operation of industry processes. In this paper, a monitoring framework jointly mining the fluid knowledge and latent property behind multi-sensor data is proposed to address the issues of uncertainty and sample scarcity in transitional flow statuses, which enables real-time monitoring on structural evolution and parameter change of flow process. Firstly, the shared iterative algorithm with multi-task weight discovery is proposed to establish the relative semantic description system to characterize flow process knowledge by mining the relationship between multi-sensor data and different fluid semantic properties, which is helpful for interpretable monitoring of flow status structure and parameter information. Secondly, latent properties mined from multi-sensor data by sparse exponential discriminant analysis contribute to flow characterization completeness and discrimination. Thirdly, the process knowledge and latent information of flow status is transformed into different degrees of describable semantic and latent properties, which obtain relative representations of flow status in the joint description space realized by rank support vector machine. The results demonstrate that the method enables precise monitoring and detailed description of typical and uncertain flow statuses in the absence of transitional flow status modeling samples.

A Data and Model-Driven Clutter Suppression Method for Airborne Bistatic Radar Based on Deep Unfolding

Space–time adaptive processing (STAP) based on sparse recovery achieves excellent clutter suppression and target detection performance, even with a limited number of available training samples. However, most of these methods face performance degradation due to grid mismatch, which impedes their application in bistatic clutter suppression. Some gridless methods, such as atomic norm minimization (ANM), can effectively address grid mismatch issues, yet they are sensitive to parameter settings and array errors. In this article, the authors propose a data and model-driven algorithm that unfolds the iterative process of atomic norm minimization into a deep network. This approach establishes a concrete and systematic link between iterative algorithms, extensively utilized in signal processing, and deep neural networks. This methodology not only addresses the challenges associated with parameter settings in traditional optimization algorithms, but also mitigates the lack of interpretability issues commonly found in deep neural networks. Moreover, due to more rational parameter settings, the proposed algorithm achieves effective clutter suppression with fewer iterations, thereby reducing computational time. Finally, extensive simulation experiments demonstrate the effectiveness of the proposed algorithm in clutter suppression for airborne bistatic radar.

Adaptive Fuzzy Fault-Tolerant Attitude Control for a Hypersonic Gliding Vehicle: A Policy-Iteration Approach

In this paper, adaptive fuzzy fault-tolerant control (AFFTC) for the attitude control system of a hypersonic gliding vehicle (HGV) experiencing an actuator fault is proposed. Actuator faults of the HGV are considered with respect to its actual structure and actuator characteristics. The HGV’s attitude system is firstly represented by a T–S fuzzy model, and then a normal T–S fuzzy controller is designed. A reinforcement learning (RL)-based policy iterative solution algorithm is proposed for the solving of the T-S fuzzy controller. Then, based on the normal T–S controller, a fuzzy FTC controller is proposed in which the control matrices can improve themselves according to the special fault. An integral reinforcement learning (IRL)-based solving algorithm is proposed to reduce the dependence of the design methods on the HGV model. Simulations on three different kinds of actuator faults show that the designed IRL-based FTC can ensure a reliable flight by the HGV.

Quadrature based innovative techniques concerning nonlinear equations having unknown multiplicity

Solution of nonlinear equations is one of the most frequently encountered issue in engineering and applied sciences. Most of the intricateed engineering problems are modeled in the frame work of nonlinear equation f(x)=0. The significance of iterative algorithms executed by computers in resolving such functions is of paramount importance and undeniable in contemporary times. If we study the simple roots and the roots having multiplicity greater of any nonlinear equations we come to the point that finding the roots of nonlinear equations having multiplicity greater than one is not trivialvia classical iterative methods. Instability or slow convergence rate is faced by these methods, and also sometimes these methods diverge. In this article, we give some innovative and robust iterative techniques for obtaining the approximate solution of nonlinear equations having multiplicity m>1. Quadrature formulas are implemented to obtain iterative techniques for finding roots of nonlinear equations having unknown multiplicity. The derived methods are the variants of modified Newton method with high order of convergence and better accuracy. The convergence criteria of the new techniques are studied by using Taylor series method. Some examples are tested for the sack of implementations of these techniques. Numerical and graphical comparison shows the performance and efficiency of these new techniques.

Optimization of Large-Scale Sparse Matrix-Vector Multiplication on Multi-GPU Systems

Sparse matrix-vector multiplication (SpMV) is one of the important kernels of many iterative algorithms for solving sparse linear systems. The limited storage and computational resources of individual GPUs restrict both the scale and speed of SpMV computing in problem-solving. As real-world engineering problems continue to increase in complexity, the imperative for collaborative execution of iterative solving algorithms across multiple GPUs is increasingly apparent. Although the multi-GPU-based SpMV takes less kernel execution time, it also introduces additional data transmission overhead, which diminishes the performance gains derived from parallelization across multi-GPUs. Based on the non-zero elements distribution characteristics of sparse matrices and the trade-off between redundant computations and data transfer overhead, this paper introduces a series of SpMV optimization techniques tailored for multi-GPU environments and effectively enhances the execution efficiency of iterative algorithms on multiple GPUs. Firstly, we propose a two-level non-zero elements-based matrix partitioning method to increase the overlap of kernel execution and data transmission. Then, considering the irregular non-zero elements distribution in sparse matrices, a long-row-aware matrix partitioning method is proposed to hide more data transmissions. Finally, an optimization using redundant and inexpensive short-row execution to exchange costly data transmission is proposed. Our experimental evaluation demonstrates that, compared with the SpMV on a single GPU, the proposed method achieves an average speedup of 2.00x and 1.85x on platforms equipped with two RTX 3090 and two Tesla V100-SXM2, respectively. The average speedup of 2.65x is achieved on a platform equipped with four Tesla V100-SXM2.

Multi-objective energy management in a renewable and EV-integrated microgrid using an iterative map-based self-adaptive crystal structure algorithm

The use of plug-in hybrid electric vehicles (PHEVs) provides a way to address energy and environmental issues. Integrating a large number of PHEVs with advanced control and storage capabilities can enhance the flexibility of the distribution grid. This study proposes an innovative energy management strategy (EMS) using an Iterative map-based self-adaptive crystal structure algorithm (SaCryStAl) specifically designed for microgrids with renewable energy sources (RESs) and PHEVs. The goal is to optimize multi-objective scheduling for a microgrid with wind turbines, micro-turbines, fuel cells, solar photovoltaic systems, and batteries to balance power and store excess energy. The aim is to minimize microgrid operating costs while considering environmental impacts. The optimization problem is framed as a multi-objective problem with nonlinear constraints, using fuzzy logic to aid decision-making. In the first scenario, the microgrid is optimized with all RESs installed within predetermined boundaries, in addition to grid connection. In the second scenario, the microgrid operates with a wind turbine at rated power. The third case study involves integrating plug-in hybrid electric vehicles (PHEVs) into the microgrid in three charging modes: coordinated, smart, and uncoordinated, utilizing standard and rated RES power. The SaCryStAl algorithm showed superior performance in operation cost, emissions, and execution time compared to traditional CryStAl and other recent optimization methods. The proposed SaCryStAl algorithm achieved optimal solutions in the first scenario for cost and emissions at 177.29 €ct and 469.92 kg, respectively, within a reasonable time frame. In the second scenario, it yielded optimal cost and emissions values of 112.02 €ct and 196.15 kg, respectively. Lastly, in the third scenario, the SaCryStAl algorithm achieves optimal cost values of 319.9301 €ct, 160.9827 €ct and 128.2815 €ct for uncoordinated charging, coordinated charging and smart charging modes respectively. Optimization results reveal that the proposed SaCryStAl outperformed other evolutionary optimization algorithms, such as differential evolution, CryStAl, Grey Wolf Optimizer, particle swarm optimization, and genetic algorithm, as confirmed through test cases.

Destination selection and flight scheduling for regional airlines at slot‐constrained airports

AbstractWe present the problem of a regional airline based at a slot‐constrained airport, which must select a set of destinations to serve, how many flights per day to operate to each destination, at what time the flights take place and which aircraft operates each flight. Restricting ourselves to the special case of regional airlines, which fly round trips from a central hub, we are able to tackle the four above decisions jointly. By contrast, in the existing literature, these decisions are usually optimised separately for generic airline networks. To solve the proposed problem, we introduce two compact integer formulations: a three‐index and a two‐index formulation. This latter, however, only solves a relaxation of the original problem because it cannot guarantee that all constraints are respected. Therefore, we embed the two‐index formulation into an iterative algorithm which dynamically adds violated constraints. Computational experiments highlight the validity of this approach and provide insights into the characteristics of the solutions.

Uncertainty quantification and data consistency analysis for the development of hydrogen and syngas oxidation model

The exponential growth of contemporary chemical kinetics data coupled with the potential inherent high uncertainty poses significant challenges for developing chemical kinetic models. To obtain a high-quality dataset for the optimization of H2 and syngas oxidation model, this study developed a framework for data consistency analysis and uncertainty analysis of chemical kinetic data. The framework combines simulations based on solving conservation equations, data-driven metaheuristic algorithms, and statistical analysis. Over 2000 experimental data points, including ignition delay times measured in shock tubes, laminar flame speeds, and concentration profiles measured in perfectly stirred reactors and plug flow reactors, were collected and used to construct the experimental dataset. Seventeen parameters of a detailed H2 and syngas oxidation model were selected to build the reaction rate constant (RRC) dataset. The initially estimated experimental uncertainties and RRC uncertainty factors were updated by slack variables in the iterative algorithm. Some high-uncertainty data were identified and excluded from the optimization dataset. The final experimental dataset achieves data consistency, and the feasible parameter set of RRC shows good prediction capability. The experimental data with high uncertainties were highlighted, and the uncertainty sources were analyzed.

Inexact Fixed-Point Proximity Algorithm for the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} Sparse Regularization Problem

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm. Specifically, the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm regularization term. Such an ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} models found by the proposed inexact algorithm outperform global minimizers of the corresponding ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document} models, in terms of approximation accuracy and sparsity of the solutions.