Nonconvex Minimization Problem Research Articles

A new algorithm for computing the nearest polynomial to multiple given polynomials via weighted ℓ2,q-norm minimization and its complex extension

A new algorithm is proposed in this paper for computing the nearest polynomial to multiple given polynomials with a given zero in the real case, where the distance between polynomials is defined by the weighted ℓ2,q norm (0<q≤2). First, the problem is formulated as a univariate constrained non-convex minimization problem, where prior information of the coefficients of the polynomial can be embedded by selecting proper weights. Then, an iteratively reweighted algorithm is designed to solve the obtained problem, and also the convergence and rate of convergence are uniformly demonstrated for all q in (0,2]. Since all the existing methods for computing the nearest polynomial to multiple given polynomials with a given zero are limited to the real case, we ingeniously extend the results to the complex case. Finally, two representative examples that separately compute the nearest real and complex polynomials are presented to show the effectiveness of the proposed algorithm.

A novel variational approach for multiphoton microscopy image restoration: from PSF estimation to 3D deconvolution

In multi-photon microscopy (MPM), a recent in-vivo fluorescence microscopy system, the task of image restoration can be decomposed into two interlinked inverse problems: firstly, the characterization of the point spread function (PSF) and subsequently, the deconvolution (i.e. deblurring) to remove the PSF effect, and reduce noise. The acquired MPM image quality is critically affected by PSF blurring and intense noise. The PSF in MPM is highly spread in 3D and is not well characterized, presenting high variability with respect to the observed objects. This makes the restoration of MPM images challenging. Common PSF estimation methods in fluorescence microscopy, including MPM, involve capturing images of sub-resolution beads, followed by quantifying the resulting ellipsoidal 3D spot. In this work, we revisit this approach, coping with its inherent limitations in terms of accuracy and practicality. We estimate the PSF from the observation of relatively large beads (approximately 1  μm in diameter). This goes through the formulation and resolution of an original non-convex minimization problem, for which we propose a proximal alternating method along with convergence guarantees. Following the PSF estimation step, we then introduce an innovative strategy to deal with the high level multiplicative noise degrading the acquisitions. We rely on a heteroscedastic noise model for which we estimate the parameters. We then solve a constrained optimization problem to restore the image, accounting for the estimated PSF and noise, while allowing a minimal hyper-parameter tuning. Theoretical guarantees are given for the restoration algorithm. These algorithmic contributions lead to an end-to-end pipeline for 3D image restoration in MPM, that we share as a publicly available Python software. We demonstrate its effectiveness through several experiments on both simulated and real data.

A nonmonotone accelerated proximal gradient method with variable stepsize strategy for nonsmooth and nonconvex minimization problems

A nonmonotone accelerated proximal gradient method with variable stepsize strategy for nonsmooth and nonconvex minimization problems

Unified Framework for Faster Clustering via Joint Schatten p-Norm Factorization With Optimal Mean.

To enhance the effectiveness and efficiency of subspace clustering in visual tasks, this work introduces a novel approach that automatically eliminates the optimal mean, which is embedded in the subspace clustering framework of low-rank representation (LRR) methods, along with the computationally factored formulation of Schatten p -norm. By addressing the issues related to meaningful computations involved in some LRR methods and overcoming biased estimation of the low-rank solver, we propose faster nonconvex subspace clustering methods through joint Schatten p -norm factorization with optimal mean (JS p NFOM), forming a unified framework for enhancing performance while reducing time consumption. The proposed approach employs tractable and scalable factor techniques, which effectively address the disadvantages of higher computational complexity, particularly when dealing with large-scale coefficient matrices. The resulting nonconvex minimization problems are reformulated and further iteratively optimized by multivariate weighting algorithms, eliminating the need for singular value decomposition (SVD) computations in the developed iteration procedures. Moreover, each subproblem can be guaranteed to obtain the closed-form solver, respectively. The theoretical analyses of convergence properties and computational complexity further support the applicability of the proposed methods in real-world scenarios. Finally, comprehensive experimental results demonstrate the effectiveness and efficiency of the proposed nonconvex clustering approaches compared to existing state-of-the-art methods on several publicly available databases. The demonstrated improvements highlight the practical significance of our work in subspace clustering tasks for visual data analysis. The source code for the proposed algorithms is publicly accessible at https://github.com/ZhangHengMin/TRANSUFFC.

Generalized and Robust Least Squares Regression.

As a simple yet effective method, least squares regression (LSR) is extensively applied for data regression and classification. Combined with sparse representation, LSR can be extended to feature selection (FS) as well, in which l1 regularization is often applied in embedded FS algorithms. However, because the loss function is in the form of squared error, LSR and its variants are sensitive to noises, which significantly degrades the effectiveness and performance of classification and FS. To cope with the problem, we propose a generalized and robust LSR (GRLSR) for classification and FS, which is made up of arbitrary concave loss function and the l2,p -norm regularization term. Meanwhile, an iterative algorithm is applied to efficiently deal with the nonconvex minimization problem, in which an additional weight to suppress the effect of noises is added to each data point. The weights can be automatically assigned according to the error of the samples. When the error is large, the value of the corresponding weight is small. It is this mechanism that allows GRLSR to reduce the impact of noises and outliers. According to the different formulations of the concave loss function, four specific methods are proposed to clarify the essence of the framework. Comprehensive experiments on corrupted datasets have proven the advantage of the proposed method.

A parameterized three-operator splitting algorithm for non-convex minimization problems with applications

A parameterized three-operator splitting algorithm for non-convex minimization problems with applications

Bregman distance regularization for nonsmooth and nonconvex optimization

Abstract Solving a nonsmooth and nonconvex minimization problem can be approached as finding a zero of a set-valued operator. With this perspective, we propose a novel Majorizer–Minimizer technique to find a local minimizer of a nonsmooth and nonconvex function and establish its convergence. Our approach leverages Bregman distances to generalize the classical quadratic regularization. By doing so, we generate a family of regularized problems that encompasses quadratic regularization as a special case. To further demonstrate the effectiveness of our method, we apply it on a lasso regression model, showcasing its performance.

Formation of wrinkles on a coated substrate using manifold‐valued finite elements

AbstractThis article treats finite element simulations of controlled wrinkle formation experiments of a soft bulk material with a thin, stiff layer on top. The wrinkling process is triggered by a stress mismatch between the bulk material and the thin layer. For the finite element simulations, we model the bulk material using a three‐dimensional hyperelastic material and the thin layer with a geometrically nonlinear elastic Cosserat shell. For the finite element simulations, we model the bulk material using a three‐dimensional hyperelastic material and the thin layer with a geometrically nonlinear elastic Cosserat shell. We use Lagrange finite elements for the bulk material and geodesic finite elements for the shell. The resulting minimization problem is nonlinear and nonconvex. We prove existence of minimizers in the continuous and the discrete function space. Finally, we solve the resulting nonconvex minimization problem numerically using a Riemannian trust‐region algorithm and compare our simulations to real experiments.

A class of modified accelerated proximal gradient methods for nonsmooth and nonconvex minimization problems

A class of modified accelerated proximal gradient methods for nonsmooth and nonconvex minimization problems

Multiobjective Optimization for Adaptive Offloading in Distributed Multiuser MIMO Cell-Free 6G Networks

By offloading some computational tasks to the edge server, edge computing can help relieve the increasing computation burden of mobile users and improve the quality of experience of users. In this paper, we investigate the computation offloading in edge computing-enabled multiuser cell-free (CF) multi-input multi-output (MIMO) networks with multiple building baseband units (BBUs). The user association, transmit covariances, CPU-cycle frequencies, and allocated bandwidths are jointly optimized to minimize two objectives: the energy consumption of mobile devices (MDs) and execution delay of tasks. We formulate it as a vector optimization problem and adopt the scalarization technique to transform it into a scalar mixed integer nonlinear programming (MINLP). Then, to solve the MINLP, we introduce user sorting into the framework of branch and bound and propose a sorted MD association (MDA) method. Two key issues are tackled in the proposed MDA, i.e., the lower bound of the MINLP and the updation of incumbent solution. For the first issue, we present a Lagrangian dual relaxation algorithm based on subgradient projection, while the second one involving another nonconvex minimization problem, we propose a successive convex approximation (SCA) based algorithm to solve it. Finally, extensive simulations demonstrate that the proposed method is able to reduce the system cost significantly and achieve better system performance by comparing with the benchmark schemes.

A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems

A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems

Tensorial Multiview Representation for Saliency Detection via Nonconvex Approach.

In the study of salient object detection, multiview features play an important role in identifying various underlying salient objects. As to current common patch-based methods, all different features are handled directly by stacking them into a high-dimensional vector to represent related image patches. These approaches ignore the correlations inhering in the original spatial structure, which may lead to the loss of certain underlying characterization such as view interaction. In this article, different from currently available approaches, a tensorial feature representation framework is developed for the salient object detection in order to better explore the complementary information of multiview features. Under the tensor framework, a tensor low-rank constraint is applied to the background to capture its intrinsic structure, a tensor group sparsity regularization is posed on the salient part, and a tensorial sliced Laplacian regularization is then introduced to enlarge the gap between the subspaces of the background and salient object. Moreover, a nonconvex tensor Log-determinant function, instead of the tensor nuclear norm, is adopted to approximate the tensor rank for effectively suppressing the confusing information resulted from underlying complex backgrounds. Further, we have deduced the closed-form solution of this nonconvex minimization problem and established a feasible algorithm whose convergence is mathematically proven. Experiments on five well-known public datasets are provided and the simulations demonstrate that our method outperforms the latest unsupervised handcrafted features-based methods in the literature. Furthermore, our model is flexible with various deep features and is competitive with the state-of-the-art approaches.

An outlier-robust smoothness-based graph learning approach

Graph learning (GL) is a tool for finding direct relationships between the nodes of a network, and hence, inferring the graph topology from the data. Recently, many GL algorithms have been proposed in the field of graph signal processing, which are based on smoothness of the graph signals on the learned graph. However, although it is possible for the input graph signals to be contaminated by outliers, for example due to sensor failures or temporary faulty information records, existing techniques are very vulnerable to outliers. So, the goal is to infer a graph topology to be, as much as possible, insensitive to this kind of data corruptions. To this aim, due to the sparse nature of outlier data, we propose a new approach for robustifying GL algorithms by incorporating L1-norm or squared L1-norm terms into the objective function of smoothness based GL methods, yielding to a non-convex minimization problem. A novel iterative minimization method is introduced to solve the resulting non-convex problem. Moreover, the convergence of the algorithm is established despite of its non-convex nature. In simulations, the high performance of the proposed algorithm is demonstrated in presence of a considerably large amount of outliers.

Adaptive Data Collection and Offloading in Multi-UAV-Assisted Maritime IoT Systems: A Deep Reinforcement Learning Approach

This paper studies the integration of data collection and offloading for maritime Internet of Things (IoT) systems with multiple unmanned aerial vehicles (UAVs). In the considered multi-UAV maritime IoT system, the UAVs act as the aerial base stations to complete the missions of data collection from buoys and data offloading to the offshore base station (OBS). In this case, the UAVs need to adaptively select the mission mode between data collection and data offloading according to the network resources and mission requirements. In this paper, we aimed to minimize the completion time of data collection and offloading missions for all UAVs by jointly optimizing the UAV trajectories, mission mode selection, transmit power of buoys, and association relationships between the UAVs and buoy/OBS. In order to solve the mixed-integer non-convex minimization problem, we first designed a multi-agent deep reinforcement learning algorithm based on a hybrid discrete and continuous action space to preliminarily obtain the UAV trajectories, mission mode selection, and the transmit power of buoys. Then, we propose an algorithm based on the stable marriage problem to determine the buoy–UAV and UAV–OBS association relationships. Finally, the simulation results show that the proposed algorithms can effectively shorten the total mission completion time of data collection and offloading for the multi-UAV-assisted maritime IoT system.

Sparse Reconstruction Using Hyperbolic Tangent as Smooth l1-Norm Approximation

In the Compressed Sensing (CS) framework, the underdetermined system of linear equation (USLE) can have infinitely many possible solutions. However, we intend to find the sparsest possible solution, which is l0-norm minimization. However, finding an l0 norm solution out of infinitely many possible solutions is NP-hard problem that becomes non-convex optimization problem. It has been a practically proven fact that l0 norm penalty can be adequately estimated by l1 norm, which recasts a non-convex minimization problem to a convex problem. However, l1 norm non-differentiable and gradient-based minimization algorithms are not applicable, due to this very reason there is a need to approximate l1 norm by its smooth approximation. Iterative shrinkage algorithms provide an efficient method to numerically minimize l1-regularized least square optimization problem. These algorithms are required to induce sparsity in their solutions to meet the CS recovery requirement. In this research article, we have developed a novel recovery method that uses hyperbolic tangent function to recover undersampled signal/images in CS framework. In our work, l1 norm and soft thresholding are both approximated with the hyperbolic tangent functions. We have also proposed the criteria to tune optimization parameters to get optimal results. The error bounds for the proposed l1 norm approximation are evaluated. To evaluate performance of our proposed method, we have utilized a dataset comprised of 1-D sparse signal, compressively sampled MR image and cardiac cine MRI. The MRI is an important imaging modality for assessing cardiac vascular function. It provides the ejection fraction and cardiac output of the heart. However, this advantage comes at the cost of a slow acquisition process. Hence, it is essential to speed up the acquisition process to take the full benefits of cardiac cine MRI. Numerical results based on performance metrics, such as Structural Similarity (SSIM), Peak Signal to Noise Ratio (PSNR) and Root Mean Square Error (RMSE) show that the proposed tangent hyperbolic based CS recovery offers a much better performance as compared to the traditional Iterative Soft Thresholding (IST) recovery methods.

A Global Dual Error Bound and Its Application to the Analysis of Linearly Constrained Nonconvex Optimization

Error bound analysis, which estimates the distance of a point to the solution set of an optimization problem4 using the optimality residual, is a powerful tool for the analysis of first-order optimization algorithms. In this paper, we use global error bound analysis to study the iteration complexity of a first-order algorithm for a linearly constrained nonconvex minimization problem. We develop a global dual error bound analysis for a regularized version of this nonconvex problem by using a novel “decomposition” technique. Equipped with this global dual error bound, we prove that a suitably designed primal-dual first-order method can generate an $\epsilon$-stationary solution of the linearly constrained nonconvex minimization problem within $\mathcal{O}(1/\epsilon^2)$ iterations, which is the best known iteration complexity for this class of nonconvex problems. The iteration complexity of our algorithm for finding an $\epsilon$-stationary solution is $\mathcal{O}(1/\epsilon^2)$, which improves the best known complexity of $\mathcal{O}(1/\epsilon^3)$ for the problem under consideration. Furthermore, when the objective function is quadratic, we establish the linear convergence of the algorithm. Our proof is based on a new potential function and a novel use of error bounds.

Determination of an Energy Source Term for Fractional Diffusion Equation

In this article, we will determine the source term of the fractional diffusion equation (FDE). Our contribution to this work is the generalization of the common inverse diffusion equation issues and the inverse diffusion equation problems for fractional diffusion equations with energy source and using Caputo fractional derivatives in time and space. The problem is reformulated in a least-squares framework, which leads to a nonconvex minimization problem, which is solved using a Tikhonov regularization. By considering the direct problem with an implicit finite difference scheme (IFDS), the numerical inversions are performed for the source term in several approximate spaces. The inversion algorithm (IA) uniqueness is obtained. Furthermore, the effect of fractional order and regularization parameter on the inversion algorithm is carried out and shows that the inversion algorithm is effective. The order of fractional derivatives expresses the global property of the direct problem and also shows the badly posed nature of the inverted problem in question.

An Adaptive Block Bregman Proximal Gradient Method for Computing Stationary States of Multicomponent Phase-Field Crystal Model

In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem's block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods.

Adaptive Bit/Power Allocation With Beamforming for Dual-Function Radar-Communication

To enhance the bit error rate (BER) performance for the dual-function radar-communication (DFRC), we investigate the problem of adaptive bit/power allocation with beamforming. Firstly, we formulate the bit/power allocation problem as a BER minimization problem subject to the radar signal-to-interference-plus-noise ratio (SINR), sidelobe information (SLI), and available resource constraints. Secondly, we devise an alternation optimization algorithm that divides the non-convex minimization problem into several low-complexity and iteratively updated subproblems. Finally, simulation results show that better communication performance, in terms of BER, can be obtained compared to state-of-the-art algorithms.

Proximal Linearized Iteratively Reweighted Algorithms for Nonconvex and Nonsmooth Optimization Problem

The nonconvex and nonsmooth optimization problem has been attracting increasing attention in recent years in image processing and machine learning research. The algorithm-based reweighted step has been widely used in many applications. In this paper, we propose a new, extended version of the iterative convex majorization–minimization method (ICMM) for solving a nonconvex and nonsmooth minimization problem, which involves famous iterative reweighted methods. To prove the convergence of the proposed algorithm, we adopt the general unified framework based on the Kurdyka–Łojasiewicz inequality. Numerical experiments validate the effectiveness of the proposed algorithm compared to the existing methods.