Search

Preserving contour topology during image segmentation is useful in many practical scenarios. By keeping the contours isomorphic, it is possible to prevent over-segmentation and under-segmentation, as well as to adhere to given topologies. The Self-repelling Snakes model (SR) is a variational model that preserves contour topology by combining a non-local repulsion term with the geodesic active contour model. The SR is traditionally solved using the additive operator splitting (AOS) scheme. In our paper, we propose an alternative solution to the SR using the Split Bregman method. Our algorithm breaks the problem down into simpler sub-problems to use lower-order evolution equations and a simple projection scheme rather than re-initialization. The sub-problems can be solved via fast Fourier transform or an approximate soft thresholding formula which maintains stability, shortening the convergence time, and reduces the memory requirement. The Split Bregman and AOS algorithms are compared theoretically and experimentally.

In this paper, we consider a class of fourth degree polynomial problems, which are NP-hard. First, we are concerned with the bi-quadratic optimization problem (Bi-QOP) over compact sets, which is proven to be equivalent to a multi-linear optimization problem (MOP) when the objective function of Bi-QOP is concave. Then, we introduce an augmented Bi-QOP (which can also be regarded as a regularized Bi-QOP) for the purpose to guarantee the concavity of the underlying objective function. Theoretically, both the augmented Bi-QOP and the original problem share the same optimal solutions when the compact sets are specified as unit spheres. By exploiting the multi-block structure of the resulting MOP, we accordingly propose a proximal alternating minimization algorithm to get an approximate optimal value of the problem under consideration. Convergence of the proposed algorithm is established under mild conditions. Finally, some preliminary computational results on synthetic datasets are reported to show the efficiency of the proposed algorithm.

This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular ${\mathscr{M}}$ -tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative componentwise nonincreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method.

Sum-of-squares (SOS) tensors plays an important role in tensor positive definiteness and polynomial optimization. So it is important to figure out what kind of tensors are SOS tensors. In this pape...

Tensor decomposition is an important research area with numerous applications in data mining and computational neuroscience. An important class of tensor decomposition is sum-of-squares (SOS) tensor decomposition. SOS tensor decomposition has a close connection with SOS polynomials, and SOS polynomials are very important in polynomial theory and polynomial optimization. In this paper, we give a detailed survey on recent advances of high-order SOS tensors and their applications. It first shows that several classes of symmetric structured tensors available in the literature have SOS decomposition in the even order symmetric case. Then, the SOS-rank for tensors with SOS decomposition and the SOS-width for SOS tensor cones are established. Further, a sharper explicit upper bound of the SOS-rank for tensors with bounded exponent is provided, and the exact SOS-width for the cone consists of all such tensors with SOS decomposition is identified. Some potential research directions in the future are also listed in this paper.

The well-known Birkhoff-von Neumann theorem states that a doubly stochastic matrix is a convex combination of permutation matrices. In this paper, we present a new concept for doubly stochastic tensors and study a generalization of this theorem for doubly stochastic tensors. Particularly, we prove that each permutation tensor is an extreme point of the set of doubly stochastic tensors, and the Birkhoff-von Neumann theorem holds for doubly stochastic tensors. Furthermore, an algorithm is proposed to find a convex combination of permutation tensors for any doubly stochastic tensor.

Level set methods are widely used in image segmentation and shape analysis. However, most of the current research focuses on fast computational algorithms, initial value selection, and practical applications in various areas. To the best of our knowledge, no research has been conducted on segmentation with level set models where the segmentation contours have to pass through some prior landmark points. In this paper, we propose a new variational model for image segmentation based on the classical Chan-Vese model for this new problem. The new model incorporates prior landmarks information as constraints in a formulated optimization problem. Then, we investigate the theoretical solvability of the new model and design a new algorithm based on the Split Bregman algorithm for numerical implementation. Finally, we conduct some segmentation experiments on gray images and compare with the original Chan-Vese model. The obtained results show many advantages of the proposed model with broad applications. Additionally, we give some critical analysis of the proposed algorithm.

It is a popular research topic in computer vision community to find a solution for the zero norm minimization problem via solving its non-convex relaxation problem. In fact, there are already many existing algorithms to solve the non-convex relaxation problem. However, most of them are computationally expensive due to the non-Lipschitz property of this problem and thus these existing algorithms are not suitable for many engineering problems with large dimensions. In this paper, we first develop an efficient algorithm to solve the non-convex relaxation problem via solving a sequence of non-convex sub-problems based on our recent work. To this end, we reformulate the minimization problem into another non-convex one but with non-negative constraint. Then we can transform the non-Lipschitz continuous non-convex problem with the non-negative constraint into a Lipschitz continuous problem, which allows us to use some efficient existing algorithms for its solution. Based on the proposed algorithm, an important relation between the solutions of relaxation problem and the original zero norm minimization problem is established from a different point of view. The results in this paper reveal two important issues: ⅰ) The solution of non-convex relaxation minimization problem converges to the solution of the original problem; ⅱ) The general non-convex relaxation problem can be solved efficiently with another reformulated high dimension problem with nonnegative constraint. Finally, some numerical results are used to demonstrate effectiveness of the proposed algorithm.

The linear complexity and the \begin{document}$ k $\end{document} -error linear complexity of a binary sequence are important security measures for the security of the key stream. By studying binary sequences with the minimum Hamming weight, a new tool, named as the hypercube theory, is developed for \begin{document}$ p^n $\end{document} -periodic binary sequences. In fact, the hypercube theory is based on a typical sequence decomposition and it is a very important tool for investigating the critical error linear complexity spectrum proposed by Etzion et al. To demonstrate the importance of hypercube theory, we first give a standard hypercube decomposition based on a well-known algorithm for computing linear complexity and show that the linear complexity of the first hypercube in the decomposition is equal to the linear complexity of the original sequence. Second, based on such decomposition, we give a complete characterization for the first decrease of the linear complexity for a \begin{document}$ p^n $\end{document} -periodic binary sequence. This significantly improves the current existing results in literature. As to the importance of the hypercube, we finally derive a counting formula for the \begin{document}$ m $\end{document} -hypercubes with the same linear complexity.

The extensive use of solar energy has promoted the development of the photovoltaic power generation industry. Traditional electric energy collection methods are easily affected by human factors, have poor real-time performance, and low reliability. The photovoltaic power generation management system based on the Internet of Things usually connects the Internet of Things equipment to a self-developed system or a third-party platform for function display and management, which can improve the above problems to a certain extent. But the lack of a unified operation and maintenance platform makes the system compatibility, scalability and stability poor. This paper designs a photovoltaic power generation management system based on NB-IoT proposes a new type of photovoltaic equipment access protocol to improve the flexibility and safety of the system and uses multiple sensors to report data analysis rules to ensure system compatibility. This establishes a data-centric, highly available photovoltaic power generation management platform to realize the unified management of different equipment manufacturers and multiple types of equipment.

Compared with deep learning methods, traditional image processing methods have lower computational costs and have advantages for typical problems such as sign text, this paper proposes a sign text location method based on pixel-level gradient ray map and edge contour mapping. First, based on image pre-processing, the pixel-level gradient ray map is obtained by using the Stroke Width Transform. Then, using the Maximally Stable Extremal Region method to locate the irregular text candidate area in the ray map. At the same time, add a mask layer to filter the irregular candidate area, and then regularize the candidate area and perform Non-Maximum Suppression processing. Finally, the background non-text candidate area is processed by extracting the mapping of the closed edge contour of the image, and the remaining text area positioning candidate frame coordinates are mapped back to the original picture. The experimental results show that the method is simple and clear, the effect is remarkable, and it has good robustness to the sign of different sizes.

In recent years, automatic detection methods based on Machine Vision have been widely used in industrial field which not only save manpower effectively but also improve product quality and detection efficiency. The energized grids produced in the industry are large in size and have a great number of connection points. Due to the poor reliability and low efficiency of traditional manual detection, a method for defect detection of energized grids based on Machine Vision is proposed. At first, the images obtained by the network camera are corrected and cropped. Then enhance the image features and extract the region of interest by image processing technology. On this basis, detect defects through the Rhombus-based Point Correction (RSPC) algorithm and the Points Rearrangement (PR) algorithm. Experimental results show that the solution proposed can detect the defects of energized grid with an accuracy rate of about 97.5% and meet the needs of industrial production.

We present a novel method for solving a complicated form of a partial differential equation called the Black-Scholes equation arising from pricing European options. The novelty of this method is that we consider two terms of the equation, namely the volatility and dividend, as variables dependent on the state price. We develop a Galerkin finite element method to solve the problem. More specifically, we discretize the system along the state variable and build new basis functions which we use to approximate the solution. We establish convergence of the proposed method and numerical results are reported to show the proposed method is accurate and efficient.

We consider multilinear systems of equations whose coefficient tensors are $${{\mathcal {M}}}$$ -tensors. Multilinear systems of equations have many applications in engineering and scientific computing, such as data mining and numerical partial differential equations. In this paper, we show that solving multilinear systems with $${{\mathcal {M}}}$$ -tensors is equivalent to solving nonlinear systems of equations where the involving functions are P-functions. Based on this result, we propose a Newton-type method to solve multilinear systems with $${{\mathcal {M}}}$$ -tensors. For a multilinear system with a nonsingular $${{\mathcal {M}}}$$ -tensor and a positive right side vector, we prove that the sequence generated by the proposed method converges to the unique solution of the multilinear system and the convergence rate is quadratic. Numerical results are reported to show that the proposed method is promising.

In this paper, we consider a class of complementarity problems with non-Lipschitzian functions arising from the reaction and diffusion problems. We develop a monotone iterative algorithm for this class of complementarity problems. The sequence generated by the proposed algorithm is monotonically decreasing and converges to the solution of the complementarity problems. Finally, numerical results are reported to demonstrate the efficiency of the proposed algorithm.

At the end of 2019, a new type of coronavirus (COVID-19) rapidly spread globally, even if the penetration of vaccination is getting higher and higher, the emergence of viral variants has increased the number of new coronal pneumonia infections. The deep learning model can help doctors quickly and accurately divide the lesion zone. However, there are many problems in the segmentation of the slice from the CT slice, including the problem of uncertainty of the disease area, low accuracy. At the same time, the semantic segmentation model of the traditional CNN architecture has natural defects, and the sensing field restrictions result in constructing the relationship between pixels and pixels, and the context information is insufficient. In order to solve the above problems, we introduced a Transformer module. Visual Transformer has been proved to effectively improve the accuracy of the model. We have designed a plug-and-play spatial attention module, on the basis of attention, increased positional offset, effective aggregate advanced features, and improve the accuracy of existing models.

As a generalization of the well-known linear complementarity problem, tensor complementarity problem (TCP) has been studied extensively in the literature from theoretical perspective. In this paper, we consider a class of TCPs equipped with nonsingular (not necessarily symmetric) M -tensors. The considered TCPs can be regarded as a special class of nonlinear complementarity problems (NCPs), but the underlying mappings in TCPs are not necessarily P 0 -functions, which preclude the possibility of applying existing Newton-type methods for NCPs to TCPs directly. By introducing a pivot minimum function, we propose an index detecting algorithm, which efficiently exploits the beneficial properties of nonsingular M -tensors and goes beyond algorithmic frameworks designed for general NCPs. Moreover, the proposed algorithm is well-defined in the sense that it generates a nonnegative element-wise nonincreasing sequence converging to a solution of the problem under consideration. Finally, numerical results further support the efficiency and reliability of the proposed algorithm.

Current studies of influence maximization focus almost exclusively on unsigned social networks ignoring the polarities of the relationships between users. Influence maximization in signed social networks containing both positive relationships (e.g., friend or like) and negative relationships (e.g., enemy or dislike) is still a challenging problem which remains much open. A few studies made use of greedy algorithms to solve the problem of positive influence or negative influence maximization in signed social networks. Although greedy algorithm is able to achieve a good approximation, it is computational expensive and not efficient enough. Aiming at this drawback, we propose an alternative method based on Simulated Annealing (SA) for the positive influence maximization problem in this paper. Additionally, we also propose two heuristics to speed up the convergence process of the proposed method. Comprehensive experiments results on three signed social network datasets, Epinions, Slashdot and Wikipedia, demonstrate that our method can yield similar or better performance than the greedy algorithms in terms of positive influence spread but run faster.

We consider approximation algorithms for nonnegative polynomial optimization problems over unit spheres. These optimization problems have wide applications e.g., in signal and image processing, high order statistics, and computer vision. Since these problems are NP-hard, we are interested in studying on approximation algorithms. In particular, we propose some polynomial-time approximation algorithms with new approximation bounds. In addition, based on these approximation algorithms, some efficient algorithms are presented and numerical results are reported to show the efficiency of our proposed algorithms.

In this paper, we propose a hybrid second-order method for homogenous polynomial optimization over the unit sphere in which the new iterate is generated by employing the second-order information of the objective function. To guarantee the convergence, we recall the shifted power method when the second-order method does not make an improvement to the objective function. As the Hessian of the objective function can easily be computed and no line search is involved in the second-order iterative step, the method is not time-consuming. Further, the new iterate is generated in a relatively larger region and thus the global maximum can be likely obtained. The given numerical experiments show the efficiency of the proposed method.

In this paper, we establish \begin{document} $Z$ \end{document} -eigenvalue inclusion theorems for general tensors, which reveal some crucial differences between \begin{document} $Z$ \end{document} -eigenvalues and \begin{document} $H$ \end{document} -eigenvalues. As an application, we obtain upper bounds for the largest \begin{document} $Z$ \end{document} -eigenvalue of a weakly symmetric nonnegative tensor, which are sharper than existing upper bounds.

M-eigenvalues of fourth-order partially symmetric tensors play an important role in nonlinear elasticity and materials. In this paper, we present some M-eigenvalue intervals to locate all M-eigenvalues of fourth-order partially symmetric tensors. It is proved that the new interval is tighter than the one proposed by He, Li and Wei [16]. Furthermore, we obtain some new checkable sufficient conditions for the strong ellipticity of fourth-order partially symmetric tensors. Three numerical examples arising from anisotropic materials are presented to verify the efficiency of the proposed results.

Consider a closed-loop supply chain including a manufacturer, a retailer and two third party recyclers as the background. A coordination and optimization model is built by using the downside-risk function, Cournot model and government subsidy excitation function. The effect of risk characteristics, government subsidy and Cournot competition on supply chain is analyzed, and the impact of revenue-and-expense sharing contract is studied in the channel, which shows that the contract cannot coordinate and optimize the closed-loop supply chain. Also, risk sharing contract and expense sharing contract are designed, which can offer the desired downside protection to the retailer, provide more profits to the agents, and accomplish channel coordination and optimization. Moreover, an application example is given for testing the effectiveness and feasibility of the contract, and the bound and rule of the contract parameters are given. Finally, by the analysis of numerical simulation and sensitivity of the model based on the contract, the validity and practicability of the model are verified, and the relationship between government subsidy, risk characteristics, competitive characteristics with the supply chain is obtained. This study provides decision support and decision-making reference for the development of remanufacturing industry.

The problem of finding a sparse solution for linear equations has been investigated extensively in recent years. This is an NP-hard combinatorial problem, and one popular method is to relax such combinatorial requirement into an approximated convex problem, which can avoid the computational complexity. Recently, it is shown that a sparser solution than the approximated convex solution can be obtained by solving its non-convex relaxation rather than by solving its convex relaxation. However, solving the non-convex relaxation is usually very costive due to the non-convexity and non-Lipschitz continuity of the original problem. This difficulty limits its applications and possible extensions. In this paper, we will consider the non-convex relaxation problem with the nonnegative constraint, which has many applications in signal processing with such reasonable requirement. First, this optimization problem is formulated and equivalently transformed into a Lipschitz continuous problem, which can be solved by many existing optimization methods. This reduces the computational complexity of the original problem significantly. Second, we solve the transformed problem by using an efficient and classical limited-memory Broyden---Fletcher---Goldfarb---Shanno algorithm. Finally, some numerical results show that the proposed method can effectively find a nonnegative sparse solution for the given linear equations with very low computational cost.

$H$-tensor is a new developed concept in tensor analysis and it is an extension of $H$-matrix and $M$-tensor. Based on the spectral theory of nonnegative tensors, several equivalent conditions of nonsingular $H$-tensors are established in the literature. However, these conditions can not be used as a criteria to identify nonsingular $H$-tensors as they are hard to verify. In this paper, based on the diagonal product dominance and $S$ diagonal product dominance of a tensor, we establish some new implementable criteria in identifying nonsingular $H$-tensors. The positive definiteness of nonsingular $H$-tensors with positive diagonal entries is also discussed in this paper. The obtained results extend the corresponding conclusions for nonsingular $H$-matrices and improve the existing results for nonsingular $H$-tensors.

In this paper, we combine the variational iteration method and perturbation theory to solve a time-dependent singularly perturbed reaction-diffusion problem. The problem is considered in the boundary layers and outer region. In the boundary layers, the problem is transformed by the variable substitution, and then the variational iteration method is employed to solve the transformed equation. In the outer region, we use the perturbation theory to obtain the approximation equation and the approximation solution. The final numerical experiments show that this method is accurate.

This paper studies the practical exponential set stabilization problem for switched nonlinear systems via a $$\tau $$?-persistent approach. In these kinds of switched systems, every autonomous subsystem has one unique equilibrium point and these subsystems' equilibria are different each other. Based on previous stability results of switched systems and a set of Gronwall---Bellman inequalities, we prove that the switched nonlinear system will reach the neighborhood of the corresponding subsystem equilibrium at every switching time. In addition, we constructively design a suitable $$\tau $$?-persistent switching law to practically exponentially set stabilize the switched system. Finally, a numerical example is presented to illustrate the obtained results.

For the nonnegative l 1 matrix/tensor sparse decomposition problem, we derive a threshold bound for the parameters beyond which all the decomposition factors are zero. The obtained result provides a guideline on selection for l 1 regularization parameters and extends the corresponding result on Lasso optimization problem.

Summary In this paper, we study the convergence property of a block improvement method (BIM) for the bi-quadratic polynomial optimization problem over unit spheres. We establish the global convergence of the method generally and establish its linear convergence rate under the second-order sufficient condition. We also extend the BIM to inhomogeneous polynomial optimization problems over unit spheres. Numerical results reported in this paper show that the method is promising. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we introduce a new portfolio selection method. Our method is innovative and flexible. An explicit solution is obtained, and the selection method allows for investors with different degree of risk aversion. The portfolio selection problem is formulated as a bi-criteria optimization problem which maximizes the expected portfolio return and minimizes the maximum individual risk of the assets in the portfolio. The efficient frontier using our method is compared with various efficient frontiers in the literature and found to be superior to others in the mean-variance space.

In this paper, the recognition of freight train's text information is studied. The text has the characteristics of discontinuous strokes, large interval between strokes and serious corrosion by environmental factors in this application scenario. The traditional template matching or geometric feature extraction cannot achieve a good recognition effect. Instead, the convolutional neural network (CNN) which is trained by a large number of image samples obtained by previous image segmentation is selected for recognition. The image segmentation processing use the Suzuki algorithm for the original image contour extraction to determine the text area. The result of contour extraction is projected horizontally using the edge information of the text area. The traversal template that combines with fixed aspect ratio of the text segmented the text image to a single text image.

Purpose – The purpose of this paper is to investigate the inverse problem of determining a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. Design/methodology/approach – The variational iteration method (VIM) is employed as a numerical technique to develop numerical solution. A numerical example is presented to illustrate the advantages of the method. Findings – Using this method, we obtain the exact solution of this problem. Whether or not there is a noisy overdetermination data, our numerical results are stable. Thus the VIM is suitable for finding the approximation solution of the problem. Originality/value – This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional and gives rapidly convergent successive approximations of the exact solution if such a solution exists.

Purpose – The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems. Design/methodology/approach – The proposed numerical technique is based on HPM to determine a heat source in the parabolic heat equation using the usual conditions. Then this shows the pertinent features of the technique in inverse problems. Findings – Using this HPM, a rapid convergent sequence which tends to the exact solution of the problem can be obtained. And the HPM does not require the discretization of the inverse problems. So HPM is a powerful and efficient technique in finding exact and approximate solutions without dispersing the inverse problems. Originality/value – The essential idea of this method is to introduce a homotopy parameter p which takes values from 0 to 1. When p=0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

We introduce M -tensors. This concept extends the concept of M -matrices. We denote Z-tensors as the tensors with nonpositive off-diagonal entries. We show that M -tensors must be Z- tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric M -tensor must be nonnegative. A symmetric M -tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an M - tensor is its smallest H + -eigenvalue and also is its smallest H-eigenvalue. We show that a Z-tensor is an M -tensor if and only if all its H + -eigenvalues are nonnegative. Some further spectral properties of M -tensors are given. We also introduce strong M -tensors, and some corresponding conclusions are given. In particular, we show that all H-eigenvalues of strong M -tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with Z-tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form.

For the problem of inaccuracy and cumulative error of visual odometer, The research and optimization of real-time Simultaneous Localization and Mapping of indoor robot based on binocular vision are studied. Based on ORB-SLAM2, key-frame map is created. First, the ORB feature is extracted from each frame of the input image and matched by fast approximation nearest neighbour(FLANN). Then, perform the preliminary pose estimation using EPnP, and optimize it with bundle adjustment and key-frame maps. When the tracking fails, apply key-frame maps and bag of words model to relocate. Finally, for the input binocular image, the SGBM is used to solve the parallax and then the depth, which will be converted to radar format data to create a map. In the research and optimization of real-time Simultaneous Localization and Mapping of indoor robot based on binocular vision, propose a method of assisted positioning with key frame map, and a method of feature matching optimization and relocation, which combines various pose optimization to achieve the accuracy of the robot indoors positioning and map construction.

SUMMARY In this paper, some important spectral characterizations of symmetric nonnegative tensors are analyzed. In particular, it is shown that a symmetric nonnegative tensor has the following properties: (i) its spectral radius is zero if and only if it is a zero tensor; (ii) it is weakly irreducible (respectively, irreducible) if and only if it has a unique positive (respectively, nonnegative) eigenvalue–eigenvector; (iii) the minimax theorem is satisfied without requiring the weak irreducibility condition; and (iv) if it is weakly reducible, then it can be decomposed into some weakly irreducible tensors. In addition, the problem of finding the largest eigenvalue of a symmetric nonnegative tensor is shown to be equivalent to finding the global solution of a convex optimization problem. Subsequently, algorithmic aspects for computing the largest eigenvalue of symmetric nonnegative tensors are discussed. Copyright © 2013 John Wiley & Sons, Ltd.

Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the largest eigenvalue for any nonnegative tensors.

In this paper, we consider the bullwhip effect problem of an Order-Up-To (OUT) inventory strategy for a supply chain system. We firstly establish a new discrete-time dynamical model which is suitable to describe the OUT inventory strategy. Then, we analyze the bullwhip effect for the dynamical model of the supply chain system. We thus transform the bullwhip effect's dampening problem to a discrete-time optimal control problem. By using the Pontryagin's maximum principle, we compute the corresponding optimal control and obtain the optimal manufacturer productivity of goods. Finally, we carry out numerical simulation experiments to show that the devised optimal control strategy is useful to dampen the bullwhip effect which always happens in the supply chain system.

In this paper, we present an iterative algorithm for computing the largest singular value of a nonnegative rectangular tensor. We establish the convergence of this algorithm for any irreducible nonnegative rectangular tensor.

In this paper we apply a recently proposed Lagrange Dual Method (LDM) to design a new Sparse Representation-based Classification (LDM-SRC) algorithm for robust face recognition problem. The proposed approach improves the efficiency of the SRC algorithm significantly. The proposed algorithm has the following advantages: (1) it employs the LDM $\ell_1$-solver to find solution of the $\ell_1$-norm minimization problem, which is much faster than other state-of-the-art $\ell_1$-solvers, e.g. $\ell_1$-magic and $\ell_1-\ell_2$. (2) The LDM $\ell_1$-solver utilizes a new Lagrange-dual reformulation of the original $\ell_1$-norm minimization problem, not only reducing the problem size when the dimension of training image data is much less than the number of training samples, but also making the dual problem become smooth and convex. Therefore it converts the non-smooth $\ell_1$-norm minimization problem into a sequence of smooth optimization problems. (3) The LDM-SRC algorithm can maintain good recognition accuracy whilst reducing the computational time dramatically. Experimental results are presented on some benchmark face databases.

We consider approximation algorithms for nonnegative polynomial optimization over unit spheres. Such optimization models have wide applications, e.g., in signal and image processing, high order statistics, and computer vision. Since polynomial functions are nonconvex, the problems under consideration are all NP-hard. In this paper, based on convex polynomial optimization relaxations, we propose polynomial-time approximation algorithms with new approximation bounds. Numerical results are reported to show the effectiveness of the proposed approximation algorithms.

In this correspondence, we propose a new Lagrange-dual reformulation associated with an l1 -norm minimization problem for sparse signal reconstruction. There are two main advantages of our proposed approach. First, the number of the variables in the reformulated optimization problem is much smaller than that in the original problem when the dimension of measurement vector is much less than the size of the original signals; Second, the new problem is smooth and convex, and hence it can be solved by many state of the art gradient-type algorithms efficiently. The efficiency and performance of the proposed algorithm are validated via theoretical analysis as well as some illustrative numerical examples.