Nonconvex Terms Research Articles

Estimation of the domain of attraction for continuous-time saturated positive polynomial fuzzy systems based on novel analysis and convexification strategies

In this paper, the domain of attraction (DOA) of the continuous-time positive polynomial fuzzy systems subject to input saturation is estimated by using the level set of the linear copositive Lyapunov function. To relax the estimation of DOA, the restriction on the level set is removed by embedding the expression of the level set into the stability conditions and positivity conditions. Referring to the nonconvex terms caused by above novel analysis strategy, some polynomial inequality lemmas are proposed to handle them; the nonconvex terms caused by imperfect premise matching (IPM) nonlinear membership functions are dealt with by sector nonlinear methods and advanced Chebyshev membership-function-dependent (MFD) methods. In this advanced MFD method, the state space segmentation and polynomial order selection of the Chebyshev approximation method are improved based on breakpoints of the first derivative and curvature, respectively, which is helpful to reduce the conservatism and computational burden of the result. Thus, this advanced Chebyshev MFD method not only optimizes the convexification strategy, but also can further be extended to estimate the DOA when it is used to introduce the membership functions information for convex stability and positivity conditions. Finally, a numerical example and the lipoprotein metabolism and potassium ion transfer nonlinear model are presented to validate the effectiveness and feasibility of the aforementioned analysis and convexification strategies in the expansion of DOA estimation.

Non-convex sparse regularization via convex optimization for blade tip timing

Blade Tip Timing (BTT), an emerging technology poised to replace strain gauges, enables contactless measurement of rotor blade vibration. However, the blade vibration signals measured by BTT systems often suffer from significant undersampling. Sparse reconstruction methods are instrumental in addressing the challenge of undersampled signal reconstruction. However, traditional approaches grounded in ℓ1 regularization tend to underestimate the amplitude of the true solution. This underestimation is particularly pronounced in the resonance state of the rotor blade, hindering effective prediction of the blade’s operational state. To overcome this limitation, this paper introduces a novel non-convex regularized BTT model, employing a non-convex penalty term with convex-preserving properties to achieve nearly unbiased reconstruction accuracy. Additionally, we propose a new threshold iteration algorithm designed for the swift solution of this model. The accuracy and robustness of the proposed method in identifying the multimodal vibration parameters of rotor blades are validated through simulations and experiments. Comparatively, the proposed method closely aligns with the Orthogonal Matching Pursuit (OMP) method in recognizing blade multimodal vibration amplitude, showcasing significant improvement over the ℓ1 regularization method. Furthermore, it demonstrates lower sensitivity to changes in BTT probe layout when compared to the OMP and ℓ1 regularization methods.

Secure beamforming design for MISO URLLC networks in IoT applications

This paper investigates joint beamforming and artificial noise (AN) design for secure multiple-input single-output (MISO) ultra-reliable and low latency communication (URLLC) networks in Internet of Things (IoT) applications. In considered system, a base station (BS) transmits confidential information for individual IoT users using the short-packet communication technique under the wiretap of eavesdroppers. To enhance physical layer security, the BS injects additional dedicated AN symbols to degrade the information retrieval ability at eavesdroppers. In this paper, we aim to joint design the transmit beamforming and AN symbol to maximize the minimum URLLC secrecy rate of IoT user subject to the power budget of the BS. The optimization design problem is highly nonconvex due to coupled variables in the URLLC secrecy rate and channel dispersion expressions, and thus it is mathematically challenging to solve this problem directly. To overcome this issue, we first introduce various convex inner approximations to convexify the nonconvex terms, and then we develop an efficient iterative algorithm based on the sequential convex programming approach. Extensive numerical simulation results are conducted to investigate the URLLC secrecy rate region. In conclusion, the two new URLLC parameters, i.e., the transmit packet blocklength and block error probability, will cause the considerable degradation on the URLLC secrecy rate region when comparing to that of the traditional beamforming design based on the Shannon capacity.

On the complexity of a quadratic regularization algorithm for minimizing nonsmooth and nonconvex functions

In this paper, we consider the problem of minimizing the function f ( x ) = g 1 ( x ) + g 2 ( x ) − h ( x ) over R n , where g 1 is a proper and lower semicontinuous function, g 2 is continuously differentiable with a Hölder continuous gradient and h is a convex function that may be nondifferentiable. This problem has important practical applications but is challenging to solve due to the presence of nonconvexities and nonsmoothness. To address this issue, we propose an algorithm based on a proximal gradient method that uses a quadratic approximation of the function g 2 and a nonconvex regularization term. We show that the number of iterations required to reach our stopping criterion is O ( max { ϵ − β + 1 β , η 2 β ϵ − 2 ( β + 1 ) β } ) . Our approach offers a promising strategy for solving this challenging optimization problem and has potential applications in various fields. Numerical examples are provided to illustrate the theoretical results.

Tensor nonconvex unified prior for tensor recovery

Tensor data, such as hyperspectral images and multi-frame videos, have gained significant attention in practical applications. However, the inherent degradation phenomena during data acquisition, including noise and missing pixels, give rise to a series of ill-posed inverse problems that need to be addressed. Currently, the rational exploration of prior knowledge for tensor recovery, including global low-rankness and local smoothness, has emerged as a common concern. Inspired by recent notable works, this paper proposes a novel tensor non-convex unified prior term, which employs weighted tensor Schatten p-norm as a rank surrogate function in the gradient domain. The new prior can yield a regularizer that effectively captures low-rankness and smoothness, and is applied to tensor completion and tensor robust principal component analysis models. An efficient algorithm is developed by using the alternating direction method of multipliers and its convergence analysis is also provided. Extensive experimental results demonstrate that the proposed method outperforms the state-of-the-art methods, particularly in cases of high missing rates and strong noise levels.

A non-convex low-rank image decomposition model via unsupervised network

Image decomposition is the separation of a given image into two parts with different features, i.e., structure and texture. In order to extract the image information comprehensively, a non-convex unsupervised image decomposition model is proposed in this paper. Firstly, we propose a new non-convex low-rank regular term to describe the low-rank nature of images, which focuses on optimizing the sum of partial singular values of image matrices using a non-convex truncated nuclear norm. Secondly, unsupervised network is used to capture image information by learning the deep semantic information of the image and then extracting the image structure. Finally, the alternative direction method of multipliers is used to divide the model into three sub-problems and optimize them separately. For the non-convex sub-problem, a two-stage solution scheme is given by employing the singular value decomposition technique, and experimental analysis is carried out on both real and synthetic images. The experimental results show that the proposed non-convex image decomposition method outperforms other state-of-the-art methods in terms of subjective visual effects. Meanwhile, in the quantitative evaluation, the proposed method shows a large improvement compared with other methods, with a maximum improvement of 15 % in the structural similarity index measure (SSIM) index.

A nonconvex sparse recovery method for DOA estimation based on the trimmed lasso

Sparse direction-of-arrival (DOA) estimation methods can be formulated as a group-sparse optimization problem. Meanwhile, sparse recovery methods based on nonconvex penalty terms have been a hot topic in recent years due to their several appealing properties. Herein, this paper studies a new nonconvex regularized approach called the trimmed lasso for DOA estimation. We define the penalty term of the trimmed lasso in the multiple measurement vector model by ℓ2,1-norm. First, we use the smooth approximation function to change the nonconvex objective function to the convex weighted problem. Next, we derive sparse recovery guarantees based on the extended Restricted Isometry Property and regularization parameter for the trimmed lasso in the multiple measurement vector problem. Our proposed method can control the desired level of sparsity of estimators exactly and give a more precise solution to the DOA estimation problem. Numerical simulations show that our proposed method overperforms traditional approaches, which is more close to the Cramer-Rao bound.

Hyperbolic tangent penalty function for edge-preserving image filtering

Edge-preserving image filtering provides a common interface for a variety of edge-aware image processing tasks. It is essential to the field of image processing and computational photography. Recent attempts at optimization models with non-convex regularization have shown promising performance in edge-preserving image filtering. In this paper, we propose a novel non-convex regularization term based on a hyperbolic tangent penalty function, which is shown to be more edge-preserving than popular non-convex penalty functions. We embed the proposed regularization term in an optimization model for edge-preserving image filtering. To solve the model with the non-convex regularization, we propose an efficient solution based on the additive half quadratic minimization and Fourier domain optimization. We have conducted extensive experiments to evaluate the proposed filter. Both quantitative and qualitative results demonstrate that our filter benefits various applications. Furthermore, the proposed filter is highly efficient and renders real-time processing of 720P color images on a modern GPU.

Physics-aware recurrent convolutional neural networks for modeling multiphase compressible flows

Multiphase compressible flow systems can exhibit unsteady and fast-transient dynamics, marked by sharp gradients and discontinuities, and material boundaries that interact with the evolving flow. The transient nature of the dynamics presents challenges to employing artificial intelligence (AI) and data-driven models for predicting flow behaviors. In this study, we explore the potential of physics-aware recurrent convolutional neural networks (PARC) to model the spatiotemporal dynamics of multiphase flows in the presence of shocks and reaction fronts. PARC is a neural network model that incorporates the generic form of the diffusion–advection–reaction equation in its network architecture, which mimics the process of solving the governing equations of fluid flows. In contrast to physics-informed machine learning approaches such as physics-informed neural networks (PINNs) where models are trained to directly minimize the residual of governing equations, PARC takes a dynamical systems viewpoint and does not seek to minimize potentially nonconvex and nonlinear loss terms. To assess the ability of PARC to accurately learn and simulate the physics of multiphase flows, we train and test PARC on various flow simulation problems, including the Burgers’ equation, fluid flow behind a cylindrical cross-section, and unsteady shock interactions with a particle at varying Mach numbers. We analyze PARC’s performance and examine sources of error in its prediction, in terms of differentiation and integration schemes and different weighting strategies for the model update. Based on our observations, we discuss PARC’s capabilities and limitations in multiphase flow applications and propose future research directions.

Positive incentive CNN structure coupled nonconvex model for image super-resolution

This paper studies super-resolution (SR) technique to reconstruct high-quality images for deep image analysis. Currently, the convolutional neural networks (CNNs) are well performing methods and the finding that random noise added in the network can have positive incentive effect, we innovatively propose a positive incentive CNNs. However, concerning the uncontrollable characteristic and lack consistency of deep network, we propose a novel framework that joins nonconvex model based on framelet and positive incentive CNN structure, which can impose consistency between the high-resolved image and the given low-resolution image, and depict image information by sparse representation. Furthermore, to overcome the challenge of computing the minimizer of the nonconvex problem, we use proximal linearized minimization (PLM) algorithm to convex the nonconvex term, then apply the alternating direction method of multipliers (ADMM) as the solver which can converge to a stationary point of the nonconvex model. The experimental outcomes on Set5, Set14, BSD100, Urban100, and real-world images demonstrate that the proposed approach outperforms the state-of-the-art methods in terms of peak signal to noise ratio (PSNR) value, structural similarity index (SSIM), and visual quality.

Impulse Noise Image Restoration Using Nonconvex Variational Model and Difference of Convex Functions Algorithm.

In this article, the problem of impulse noise image restoration is investigated. A typical way to eliminate impulse noise is to use an L1 norm data fitting term and a total variation (TV) regularization. However, a convex optimization method designed in this way always yields staircase artifacts. In addition, the L1 norm fitting term tends to penalize corrupted and noise-free data equally, and is not robust to impulse noise. In order to seek a solution of high recovery quality, we propose a new variational model that integrates the nonconvex data fitting term and the nonconvex TV regularization. The usage of the nonconvex TV regularizer helps to eliminate the staircase artifacts. Moreover, the nonconvex fidelity term can detect impulse noise effectively in the way that it is enforced when the observed data is slightly corrupted, while is less enforced for the severely corrupted pixels. A novel difference of convex functions algorithm is also developed to solve the variational model. Using the variational method, we prove that the sequence generated by the proposed algorithm converges to a stationary point of the nonconvex objective function. Experimental results show that our proposed algorithm is efficient and compares favorably with state-of-the-art methods.

Regularized Discrete Optimal Transport for Class-Imbalanced Classifications

Imbalanced class data are commonly observed in pattern analysis, machine learning, and various real-world applications. Conventional approaches often resort to resampling techniques in order to address the imbalance, which inevitably alter the original data distribution. This paper proposes a novel classification method that leverages optimal transport for handling imbalanced data. Specifically, we establish a transport plan between training and testing data without modifying the original data distribution, drawing upon the principles of optimal transport theory. Additionally, we introduce a non-convex interclass regularization term to establish connections between testing samples and training samples with the same class labels. This regularization term forms the basis of a regularized discrete optimal transport model, which is employed to address imbalanced classification scenarios. Subsequently, in line with the concept of maximum minimization, a maximum minimization algorithm is introduced for regularized discrete optimal transport. Subsequent experiments on 17 Keel datasets with varying levels of imbalance demonstrate the superior performance of the proposed approach compared to 11 other widely used techniques for class-imbalanced classification. Additionally, the application of the proposed approach to water quality evaluation confirms its effectiveness.

An initially robust minimum simplex volume-based method for linear hyperspectral unmixing

ABSTRACT Initialization plays an important role in the accuracy of endmember extraction algorithms (EEAs) in linear hyperspectral unmixing (LHU). Random initialization can lead to varying endmembers generated by EEAs. To address this challenge, an initialization strategy has been introduced, encompassing vertex component analysis (VCA), automatic target generation process (ATGP), among others. These techniques significantly contribute to enhancing the accuracy of EEAs. However, complex initialization is sometimes less preferable, prompting the unexplored question of whether there exists an EEA robust to initialization. This paper focuses on analyzing this issue within the context of minimum simplex volume-based (MV) methods, which have received considerable attention in the past two decades due to their robustness against the absence of pure pixels. MV methods typically formulate LHU as an optimization problem, most of which includes a non-convex volume term. Additionally, many MV methods use VCA as an initialization strategy. Firstly, this paper demonstrates that the variable splitting augmented Lagrangian approach (SISAL), as a representative non-convex MV method, heavily depends on initialization. To our knowledge, the impact of initialization for MV methods has not been thoroughly analyzed before. Furthermore, this paper proposes an initially robust MV method by introducing a new convex MV term. Numerical experiments conducted on simulated and real datasets demonstrate its outstanding performance in accuracy and robustness to initialization. Throughout the experiments the proposed method proves to be the most stable, which is crucial in real scene where the ground truth is unknown beforehand.

Nonconvex fractional order total variation based image denoising model under mixed stripe and Gaussian noise

In this paper, we propose a minimization-based image denoising model for the removal of mixed stripe and Gaussian noise. The objective function includes the prior information from both the stripe noise and image. Specifically, we adopted a unidirectional regularization term and a nonconvex group sparsity term for the stripe noise component, while we utilized a nonconvex fractional order total variation (FTV) regularization for the image component. The priors for stripes enable adequate extraction of periodic or non-periodic stripes from an image in the presence of high levels of Gaussian noise. Moreover, the nonconvex FTV facilitates image restoration with less staircase artifacts and well-preserved edges and textures. To solve the nonconvex problem, we employed an iteratively reweighted $ \ell_1 $ algorithm, and then the alternating direction method of multipliers was adopted for solving subproblems. This led to an efficient iterative algorithm, and its global convergence was proven. Numerical results show that the proposed model provides better denoising performance than existing models with respect to visual features and image quality evaluations.

Performance analysis of the convex non-convex total variation denoising model

<p>Total variation (TV) regularization is a powerful tool in image denoising, but it often exhibits limited performance in preserving edges. In contrast, non-convex TV regularization can more effectively preserve edges and contours, albeit posing challenges when solving. Recently, the convex non-convex (CNC) strategy has emerged as a potent approach that allows incorporating non-convex TV regularization terms while maintaining the overall convexity of the objective function. Its superior performance has been validated through various numerical experiments; however, theoretical analysis remains lacking. In this paper, we provided theoretical analysis of the performance of the CNC-TV denoising model. By utilizing the oracle inequality, we derived an improved upper bound on its performance compared to TV regularization. In addition, we devised an alternating direction method of multipliers (ADMM) algorithm to address the proposed model and verified its convergence properties. Our proposed model has been validated through numerical experiments in 1D and 2D denoising, demonstrating its exceptional performance.</p>

Trajectory planning for satellite swarms with nonlinear terminal constraints using penalty concave relaxation

Convex programming (CP) is widely used in trajectory planning for space missions due to its high efficiency. Existing CP methods may not handle well with trajectory planning problem for satellite swarms involving multiple nonlinear terminal constraints, leading to solution chattering and initial infeasibility. To overcome this issue, this paper introduces a penalty concave relaxation (PCR) treatment to the conventional CP methods. First, the nonconvex terms in multiple nonlinear terminal constraints are equivalently concentrated into a unique equality constraint to reduce the number of relaxation variables. This equality constraint is then relaxed along the concave direction to guarantee the initial feasibility of the subsequent iterative solution process. Meanwhile, only one penalty coefficient is added on the objective. Finally, the relaxed concave constraint is linearized so that chattering can be avoided in an iterative strategy to solve the original swarm trajectory planning problem. Numerical examples are set up for two swarm trajectory planning missions with nonlinear terminal constraints. The performance of avoiding chattering and ensuring initial feasibility is verified.

Proximal linearized method for sparse equity portfolio optimization with minimum transaction cost

In this paper, we propose a sparse equity portfolio optimization model that aims at minimizing transaction cost by avoiding small investments while promoting diversification to help mitigate the volatility in the portfolio. The former is achieved by including the ell _{0}-norm regularization of the asset weights to promote sparsity. Subjected to a minimum expected return, the proposed model turns out to be an objective function consisting of discontinuous and nonconvex terms. The complexity of the model calls for proximal method, which allows us to handle the objective terms separately via the corresponding proximal operators. We develop an efficient algorithm to find the optimal portfolio and prove its global convergence. The efficiency of the algorithm is demonstrated using real stock data and the model is promising in portfolio selection in terms of generating higher expected return while maintaining good level of sparsity, and thus minimizing transaction cost.

Energy scaling laws for microstructures: from helimagnets to martensites

We consider scalar-valued variational models for pattern formation in helimagnetic compounds and in shape memory alloys. Precisely, we consider a non-convex multi-well bulk energy term on the unit square, which favors four gradients (±α,±β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\pm }\\,\\alpha ,{\\pm }\\,\\beta )$$\\end{document}, regularized by a singular perturbation in terms of the total variation of the second derivative. We derive scaling laws for the minimal energy in the case of incompatible affine boundary conditions in terms of the singular perturbation parameter as well as the ratio α/β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha /\\beta $$\\end{document} and the incompatibility of the boundary condition. We discuss how well-studied models for martensitic microstructures in shape-memory alloys arise as a limiting case, and relations between the different models in terms of scaling laws. In particular, we show that scaling regimes arise in which an interpolation between the rather different branching-type constructions in the spirit of Kohn and Müller (Commun Pure Appl Math 47(4):405–435, 1994) and Ginster and Zwicknagl (J Nonlinear Sci 33:20, 2023), respectively, occurs. A particular technical difficulty in the lower bounds arises from the fact that the energy scalings involve various logarithmic terms that we capture in matching upper and lower scaling bounds.

Generalized efficient robust predictive control for networked interval type-2 T–S fuzzy system with adaptive event-triggered scheme

This article studies the generalized efficient robust predictive control (GERPC) for the nonlinear systems denoted by interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy model over networks with consideration of packet dropout, bounded disturbance and adaptive event-triggered scheme (AETS). First, the packet dropout is described by a Bernoulli process, and an AETS with thresholds that can be self-regulated according to different situations is employed to decide whether to release data into the communication network, so as to reduce the data transmission frequency and the network burden; Second, taking into consideration of packet dropout and AETS, a unified IT2 T–S fuzzy control model based on GERPC is established, and the stability of closed-loop system with bounded disturbance is guaranteed by quadratic boundedness (QB) technique; Third, generalized efficient robust predictive controller is designed to obtain a larger initial feasible region, smaller online computation and acceptable control performance. The unconstrained control law is designed offline, and a new variable will be optimized online to achieve good control performance, which is introduced to adjust the control quantity. Then, the convex LMI reformulation method is utilized to handle non-convex terms in optimization problem, and the effectiveness of the GERPC algorithm is proved by the matrix partition technique. At last, the effectiveness of the designed controller based on the GERPC strategy is verified by a given simulation experiment.

Efficient waveform design with jamming characteristics for precision electronic warfare

The goal of precision electronic warfare (PREW) is to implement blanket jamming on the targets precisely and ensure that friendly devices are not affected; thus, it requires controlling energy precisely in the spatial domain and covering the working bandwidth of the target equipment in the frequency domain. In this paper, we propose the algorithms to design waveforms with jamming characteristics for PREW, where the frequency requirement is converted to mitigation on the autocorrelation peak sidelobe level (APSL) of the combined waveform in the time domain. Then, we establish two multiobjective problems (MOPs) based on one-step and two-step methods, which have their own advantages in practice; that is, the former performs better on the computational efficiency by obtaining the waveform directly, and the latter controls the energy better in the worst-case scenario by recovering the covariance matrix of the waveform. To solve the two MOPs, the ℓp-norm and a reasonable conversion are employed to approximate the highly nonconvex APSL terms, and the objective functions are transformed into quadratic terms within the majorization-minimization (MM) framework. An accelerating scheme is applied to expedite the proposed algorithms. Numerical examples demonstrate that the two proposed algorithms outperform the existing algorithms on the jamming performance and computational burden.