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Abstract: In this paper, we will derive a solver for a symmetric strongly nonsingular higher order generator representable semiseparable plus band matrix. The solver we will derive is based on the Levinson algorithm, which is used for solving strongly nonsingular Toeplitz systems. In the first part an solver for a semiseparable matrix of semiseparability rank p is derived, and in a second part we derive an solver for a band matrix with bandwidth . Both solvers are constructed in a similar way: firstly a Yule–Walker-like equation needs to be solved, and secondly this solution is used for solving a linear equation with an arbitrary right-hand side. Finally, a combination of the above methods is presented to solve linear systems with semiseparable plus band coefficient matrices. The overall complexity of this solver is plus lower order terms. In the final section numerical experiments are performed. Attention is paid to the timing and the accuracy of the described methods. [Copyright &y& Elsevier]

System identification determines the model of the plant from the measurement data and has been widely used in the prediction and control. In this paper, the parameter estimation problem is studied for multivariate equation-error systems with autoregressive moving average noises. Since the considered system is high-dimensional, the distributed identification is considered in this paper. The subsystems are obtained by decomposing the original system in accordance with the number of the outputs. However, the subsystems contain the same parameter vector, resulting in many redundant estimates. By taking the average of the parameter estimation vectors, we develop a partially-coupled subsystem recursive generalized extended least squares (PC-S-RGELS) algorithm to cut down the redundant parameter estimates. In consideration of the information communication between the subsystems and the convergence of the estimation algorithm, a partially-coupled RGELS (PC-RGELS) algorithm is presented by means of the coupling identification concept. Compared with the multivariate RGELS algorithm, the two algorithms have higher computational efficiencies. Finally, an illustrative example is provided to demonstrate the effectiveness of the two proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, an accelerated spectral conjugate gradient projection algorithm is proposed for solving the constrained nonlinear pseudo-monotone equations. We set a restart procedure related to the conjugate parameter in the search direction of the spectral conjugate gradient method. We use an accelerated gradient descent scheme to generate the spectral parameter. Finally, we adopt the relaxed-inertial strategy to accelerated algorithm iteration process. Global convergence is proved under mild conditions without the Lipschitz continuity assumption. Numerical comparisons with other methods show the performances of our algorithm for solving large-scale equations. Our applications are on regularized decentralized logistic regression and image de-blurring problems. [ABSTRACT FROM AUTHOR]

The outbreak of uncertain events, e.g., financial crisis, regional conflict and abrupt contagion, has a significant impact on residents' income. Hence, the wealth management and portfolio selection become more and more important. In addition, the behavioral finance believes that decision-making process of the investors not only depend on utility maximization, but also on who to compare with. It differs from the traditional finance based on rational cognitive acquiring and decision-making of agents, in this paper, we consider both investment uncertainty and investment utility with reference level for portfolio selection. First, we introduce the uncertainty theory and loss aversion utility with reference level based on behavioral finance. Then we study an uncertain multi-period portfolio selection problem. Moreover, we formulate an uncertain behavioral portfolio selection model with three objective functions for which we propose an equivalent program consisting of a main model and two auxiliary models for tri-objective optimization model. Thereafter, we use a modified evolutionary root system growth algorithm to solve the transformed models. Finally, a numerical simulation allows to show the practicability of proposed models and validity of modified algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we first summarize the algorithm presented in Blasco and Pérez-Díaz (2014) for computing the generalized asymptotes of algebraic curves implicitly defined. This algorithm is based on the computation of Puiseux series. The main and very important contribution of this paper is a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic plane curve implicitly defined by just solving a triangular system of equations. The method can be easily generalized to the case of algebraic curves implicitly defined in the n -dimensional space. [ABSTRACT FROM AUTHOR]

The Peaceman–Rachford splitting method (PRSM) has been widely studied in recent years because of its excellent numerical performance, especially with inertial acceleration. In this paper, we propose a multi-step inertial strictly contractive PRSM (shortly, MISCPRSM) for solving convex optimization problems, which has not been studied before. This paper aims to give an exhaustive analysis of the value relationship between multiple inertial parameters. The second subproblem utilizes an additional customized indefinite proximal term in order to obtain a closed-form solution. The global convergence and complexity result of the introduced method are analyzed by using variational inequality and basic inequalities. Finally, through simulation and computed tomography experiments, the effectiveness and robustness of the MISCPRSM are proved, and the detailed values of inertial parameters are given. [ABSTRACT FROM AUTHOR]

This paper presents a uniform restricted isometry property (RIP) analysis of the ℓ p − ω ℓ q minimization method for the recovery of sparse signals in both noiseless and noisy scenarios. The analysis is primarily based on a crucial sparse representation for any given signals. Moreover, we propose an effective algorithm for solving the ℓ p − ω ℓ q minimization problem. The performance of this algorithm is validated through extensive numerical experiments on synthetic data, ECG signals, and grayscale images. [ABSTRACT FROM AUTHOR]

In this paper, we put forward a new structure-preserving algorithm based on the semi-tensor product of matrices for the split quaternion matrix LDU decomposition, which makes full use of high-level operations, and relation of operations between split quaternion matrices and their L -representation matrices. The algorithm is more stable and efficient than the complex structure-preserving algorithm proposed in Wang et al. (2021). Numerical experiments are provided to verify the efficiency of the new structure-preserving algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the sparse solution of the tensor complementarity problem (TCP), which is an NP hard problem due to the nonconvexity and noncontinuity of the ℓ 0 norm. We transform the complementarity constraints into a fixed point equation with projection type and use 0/1-loss function to relax ℓ 0 norm. We develop a smoothing Newton projection (SNP) algorithm to solve the transformed problem. The subproblem with 0/1-loss function is solved by Newton method. Finally, numerical results show that the SNP algorithm can effectively get the sparse solution to a TCP. [ABSTRACT FROM AUTHOR]

In this paper, a new fully discrete variational Crank–Nicolson ensemble (FVE/CN) scheme is established for the heat equation with uncertainty, then a fully discrete variational Crank–Nicolson ensemble Monte Carlo algorithm (FVEMC/CN) is given. Using the FVEMC/CN algorithm, one just needs to solve a single linear system with multiple right-hand side vectors in a group at each time step. The stability and the error estimates are presented to indicate the proposed algorithm is second-order convergent regarding the time. Finally, numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed approaches. [ABSTRACT FROM AUTHOR]

This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring–Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak–Ribière–Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis. [ABSTRACT FROM AUTHOR]

The present paper aims to regularize ill-posed high-dimensional problems using a constrained total variation minimization approach. The analysis of the continuous model via the dual formulation converts this minimization problem into a constrained minimax problem. Three main issues stand before the resolution of this problem: The minimization over a constraint, the high dimensional representation of the model, and the non-linearity and non-differentiability of the cost function. On the one hand, multidimensional data analysis will be performed using the tensor representation. On the other hand, a new alternating approach based on a proposed pseudo-Lagrangian operator will be developed to deal with the non-linearity and the non-differentiability of this constrained optimization problem. The proposed technique aims to decompose the main tensorial problem into feasible subproblems that can be solved using the conditional gradient method. The convergence of the proposed algorithm is proved and some numerical results are given to illustrate the effectiveness of the presented approach in different fields of image and video processing. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new self-adaptive algorithm for finding a solution to the system consisting of pseudomonotone variational inequalities and split fixed point problems. Our algorithm does not depend on the norm of the transfer mapping and the Lipschitz constant of the operators. In addition, the control parameters are self-adaptive. We also give some numerical examples to illustrate the effectiveness of the proposed algorithm and compare it with Lohawech et al.'s algorithm in Lohawech et al. (2018) and Nnakwe et al.'s algorithm in Nnakwe et al. (2021). [ABSTRACT FROM AUTHOR]

Parameter-free filled functions have become a new direction for the auxiliary function approach development as parameters serve as the main barrier of the filled function's efficiency. However, the parameter-free filled function suffers from at least three shortcomings, namely, the use of an exponential function, a lower semi-continuous property, and the fulfillment of the third axiom of the filled function definition. This paper intends to address these limitations by providing a new inflection point-based auxiliary function. This function has continuously differentiable and non-exponential properties. To show the competitiveness of the proposed method, we conduct a comparison with some recently introduced filled function algorithms. Numerical results show the superiority of the proposed method. • The fulfillment of the third axiom of the filling properties is still a challenge. • The third axiom can be weakened to the existence of the inflection point. • Numerical simulation shows that the inflection-based filled function is efficient. [ABSTRACT FROM AUTHOR]

In this paper, we realize a rapid, smooth iterative morphing algorithm between two immersed and closed spherical curves of arbitrary turning numbers. By compensating in turning numbers, we linearly interpolate between geodesic curvatures and the side lengths of their accurate spherical polygons approximation. Our algorithm is essentially based on a geometrical closure condition for a flow of spherical polygons involving the variation of both sides and exterior angles. A curvature smoothing flow using discrete geodesic curvature is adopted for smooth curve generation. To demonstrate the good properties of the method, a morphing of spherical curves and ruled surfaces is illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the two-block nonconvex and nonsmooth optimization with linear constraints, where the objective function is the sum of a convex but nonsmooth function and a smooth but nonconvex function. This problem encompasses many important applications in engineering and machine learning. Based on the splitting algorithm, making use of the quadratic approximation of the smooth part, and with the help of Armijo line search technique, we first propose a splitting augmented Lagrangian method with partial quadratic approximation. Under some mild conditions, the global convergence of the proposed method is proved. Moreover, when the gradient of the smooth part in the objective function is Lipschitz continuous, we then propose and analyze a variant of the aforementioned method without performing line search. We report some preliminary numerical results on solving nonconvex quadratic regularization problems to show the feasibility and effectiveness of the two proposed methods. Finally, we show applicability and encouraging efficiency of our methods by applying them to solve sparse signal recovery problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient trust region algorithm for unconstrained optimization problem. Different from the traditional trust region algorithm, our algorithm obtains the next iteration point by projection after getting the trust region trial step. The main feature of this algorithm is that it generates a bounded iteration sequence automatically, and it retains the excellent convergence properties. The numerical tests for support vector machine(SVM) and some unconstrained optimization test sets problems show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, an Anderson's acceleration algorithm for the finite element Oseen iterative scheme of the steady incompressible magnetohydrodynamics (MHD) problem is designed and analyzed. The Anderson acceleration algorithm based on solving an optimization problem in each iteration receives great attention to accelerate linear and nonlinear iterations. We prove that the proposed algorithm has higher convergence rate than that of the Oseen iterative method for the steady incompressible MHD, and give the accelerating convergence theory under different acceleration depths m. Numerical experiments including the singular solution and benchmark problems are carried out to verify the correctness and effectiveness of the algorithm. The tests show the marked improvement in the high physical parameters (including the hydrodynamic and magnetic Reynolds numbers, and the coupling number) when the Oseen iteration fails to converge. [ABSTRACT FROM AUTHOR]

The Barzilai–Borwein (BB) method is a popular and efficient gradient method for solving large-scale unconstrained optimization problems. In general, it converges much faster than the steepest descent (Cauchy) method. However, it may not converge, even when the objective function is strongly convex. To overcome this, by virtue of bounding the distance between sequential iterates, [Burdakov et al. (2019)] proposed a stabilized BB method. In this paper, by combining the stabilized BB method with a projection approach, we develop a stabilized BB projection (SBBP) method to solve nonlinear monotone equations with convex constraints. SBBP does not involve computing matrices proving the usefulness for solving large-scale problems. Its global convergence and Q-linear convergence rate are also established. We report numerical experiments on recovering sparse signals and restoring blurred images arising from compressive sensing, demonstrating the usefulness of SBBP. [ABSTRACT FROM AUTHOR]

This paper investigates a type of minimax linear fractional program (MLFP) that often occurs in practical problems such as design of electronic circuits, finance and investment. We first transform the MLFP problem into an equivalent problem (EP) by using the Charnes–Cooper transformation and introducing an auxiliary variable. A linear relaxation strategy should simplify the nonconvex parts of the constraints in (EP). For globally solving MLFP, a branch-and-bound algorithm is then developed. It integrates the presented relaxation with the one-dimensional branching. The convergence of the algorithm is demonstrated and the number of worst-case iterations is estimated. Finally, preliminary numerical experiments verify that the proposed algorithm in this article is robust and efficient for solving the tested instances. [ABSTRACT FROM AUTHOR]

Multi-objective programming problem often contains numerous efficient solutions, which confuses the decision-maker. To assist in selecting the most desirable solution, optimizing a function over the efficient set becomes crucial. In this paper, we present a novel method for optimizing a general quadratic function over the efficient set of a multi-objective integer linear programming problem. To solve this problem, a ranking approach and efficiency test is utilized. The proposed methodology obtains a globally optimal solution by systematically scanning ranked solutions of an integer quadratic programming problem until the efficiency condition is satisfied. For generating ranked solutions, we construct a related integer linear programming problem. Then, ranked solutions of the integer linear programming problem are used for enumerating ranked solutions of the integer quadratic programming problem. The convergence of our algorithm is established theoretically, and its steps are illustrated using a numerical example. Aparticular case of the proposed method for optimizing a linear function over the efficient set of a multi-objective integer linear programming problem is also discussed. Further, extensive computational results demonstrate the effectiveness of our method for solving problems with large number of constraints, variables, and objective functions. Moreover, comparative analysis shows that the developed algorithm came out to be computationally more efficient as compared to the existing state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

In this paper, non-overlapping domain decomposition algorithms for the Stokes problem with a discontinuous viscosity coefficient are proposed and analyzed. A pair of inf–sup stable finite element spaces with discontinuous pressure finite element functions is employed to obtain a discrete Stokes system. By introducing a non-overlapping subdomain partition for the given finite element spaces, the discrete Stokes system is reformulated into an algebraic system on dual unknowns, where the continuity on the velocity unknowns at the subdomain vertices is enforced strongly and the continuity on the remaining interface velocity unknowns is enforced weakly using the dual unknowns. The resulting algebraic system on the dual unknowns is then solved by an iterative method. The coarse problem in the resulting algebraic system is only related to the strongly coupled primal velocity unknowns and it is thus obtained as a symmetric positive definite matrix equation. To accelerate the iteration convergence, practical versions of lumped and Dirichlet preconditioners are proposed and their condition numbers are also analyzed for the resulting dual algebraic system. Numerical results are also included to confirm our condition number estimates. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce new iterative algorithms for approximating a solution to a class of monotone operator equations. More precisely, we study the split common solution problem with multiple output sets for monotone operator equations in Hilbert spaces. In order to solve this problem, we propose three new algorithms and establish strong convergence theorems for them. [ABSTRACT FROM AUTHOR]

We study a multi-block separable convex optimization problem with the linear constraints, where the objective function is the sum of m individual convex functions without overlapping variables. The linearized version of the generalized alternating direction method of multipliers (L-GADMM) is particularly efficient for the two-block separable convex programming problem and its convergence was proved when two blocks of variables are alternatively updated. However, the convergence analysis and practical applications of the extended L-GADMM (m ≥ 3) are in their early stages. In this paper, we extend this algorithm to the general case where the objective function consists of the sum of m -block convex functions. Theoretically, we prove global convergence of the new method and establish the worst-case convergence rate in the ergodic and nonergodic senses for the proposed algorithm. The efficiency of the new method is further demonstrated through numerical results on the calibration of the correlation matrices. [ABSTRACT FROM AUTHOR]

Abstract This paper suggests a novel continuous-time gradient descent algorithm of a constrained convex unknown function with a stochastic noise in the observed data. The penalty function approach is employed to introduce the restrictions of the system and to provide a successful optimization process. The solution is restricted to a static scheme dealing with the class of strongly convex functions subject to a set of constraints. To estimate the stochastic gradient we employ a modified version of the synchronous detection method. All the parameters of the proposed approach are decreasing in time to both compensate noise effect in the observations and to provide the mean-square convergence of the suggested approach. We present two different numerical examples to validate the contributions of the paper. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new viscosity iterative method minimization algorithm for image restoration algorithm is proposed. Based on a new viscosity iteration, the algorithm for finding the common zeros of two accretive operators in the framework of uniformly smooth Banach spaces. Moreover, the strong convergence theorems for the iterative algorithms and an example is proposed which shows the validity of main theorem are proved. The results of this paper are improved and extended of the corresponding ones announced by many others and we also applied our result to solve a convex minimization problem and Gâteaux differentiable E P s and Variational Inequality. Experiment results show that the proposed algorithms outperform some other methods. Finally, we give the numerical experiments to show the efficiency and advantage of the proposed methods and we also used our proposed algorithm for applying to solve the image deblurring and image recovery problems. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a simplified reproducing kernel method to solve the linear Volterra integral equations with variable coefficients. The main idea of the method is to establish a reproducing kernel direct space that can be used in Volterra integral equations. And in the first time, this paper analyzes the convergence order and stability of the approximate solution. Then the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The proposed method is proved to be stable and is not less than the second order convergence. The algorithm is proved to be feasible and stable through some numerical examples. [ABSTRACT FROM AUTHOR]

Box-constrained optimization problems in the real m × n matrix space have been widely applied in big data mining. However, efficient solution of them is still a challenge. In this paper, a new nonmonotone line search rule is first proposed by extending the well-known ones and inheriting their advantages. Then, by analyzing and exploiting properties of this rule, a new nonmonotone spectral projected gradient algorithm is developed to solve the box-constrained optimization problems in the matrix space. Global convergence of the developed algorithm is also established. Numerical tests are conducted on a series of randomly generated test problems and those in the set of benchmark test problems. Compared with other existing nonmonotone line search rules, our rule shows its advantages in terms of the significantly reduced number of function evaluations and significantly reduced number of iterations. To further validate applicability of this research, we apply the studied optimization problem and the developed algorithm to solve the problems of image clustering. Numerical results demonstrate that the proposed method can generate better clustering results and is more robust than the similar ones available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for block sparse signal recovery that minimizes the block q -ratio sparsity ‖ z ‖ 2 , 1 / ‖ z ‖ 2 , q q q − 1 with q ∈ [ 0 , ∞ ]. For the case of 1 < q ≤ ∞ , we present the theoretical analyses and the computing algorithms for both cases of the ℓ 2 -bounded and ℓ 2 , ∞ -bounded noises. The corresponding unconstrained model is also investigated. Its superior performance in block sparse signal reconstruction is demonstrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a new descent approximate modified residual algorithm is developed to solve a large scale system of nonlinear symmetric equations, where the basic strategy to improve its numerical performance is to approximately compute the gradients and the difference of gradients. The error bounds of this approximation are presented, and in virtue of this approximation, a conjugate gradient algorithm for solving large-scale optimization problems in the literature is extended to solve the system of nonlinear symmetric equations without needs of computing and storing the Jacobian matrices or their approximate matrices. It is proved that the obtained search directions in our developed algorithm are sufficiently decent with respect to the so-called approximate modified residues. Under mild assumptions, global and local convergence results of the developed algorithm are proved. Numerical tests indicate that the developed algorithm outperforms the other similar ones available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of interval-valued multiobjective optimization problems (IVMOP), and formulate a certain equivalent multiobjective optimization problem (MOP). We establish that any Pareto optimal solution of the equivalent MOP is also an effective solution of IVMOP. We introduce two variants of the quasi-Newton method to solve IVMOP when every component of the objective function has second-order partial generalized Hukuhara derivatives. In the first variant, we assume that the objective function of IVMOP is strongly convex, and approximate Hessian matrix of each component of the objective function of MOP by using the well-known BFGS method. The algorithm generated by this variant exhibits superlinear convergence to a locally effective solution of IVMOP. In the second variant, we approximate the Hessian matrix of each component of MOP by applying the Levenberg–Marquardt method. Moreover, this variant can be applied to non-convex functions, and the sequence generated by this variant converges to a locally effective solution of IVMOP as well. Finally, some numerical examples are furnished to illustrate the effectiveness of our developed methodology. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a generalized version of proximal method with a new step size update for solving variational inequality problem in Hilbert spaces. Weak convergence of the proposed scheme is attained under some appropriate assumptions imposed on the operators and parameters. At the same time, we set up an R-linear convergence rate of our method when the operator is strong monotone. Numerical experiments are given to demonstrate the efficacy of our proposed iterative algorithm. Several versions of recently proposed methods in the literature are special cases of our method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two new self-adaptive iterative algorithms for finding the solution of a split variational inclusion problem in real Hilbert spaces. The strong convergence theorems are proved under suitable conditions. In applications, the main results are applied for solving the split feasibility problem, the split minimization problem, and the split common fixed point problem. In order to test the numerical performances, the proposed algorithms are compared with several existing algorithms in the literature. Finally, the computational experiments show that the proposed algorithms are fast and more efficient than other approaches. [ABSTRACT FROM AUTHOR]

The use of decision rules allows to extract information and to infer conclusions from relational databases in a reliable way, thanks to some indicators like support and certainty. Moreover, decision algorithms collect a group of decision rules that satisfies desirable properties to describe the relational system. However, when a decision table is considered within a fuzzy environment, it is necessary to extend all notions related to decision algorithms to this framework. This paper presents a generalization of these notions, highlighting the new definitions of indicators of relevance to describe decision rules and decision algorithms. [ABSTRACT FROM AUTHOR]

Finite tight frames are used widely for many applications. An important problem is to construct finite tight frames containing specified elements as needed in real world applications. The question was first studied in details by D. Feng, L. Wang, and Y. Wang in 2007. P. G. Casazza, M. Ficus, D. G. Mixon, J. Peterson, and I. Smalyanau gave a method for constructing a tight frame containing specified elements in 2013. A simple and stable method for constructing equi-norm tight frames from the specified elements was derived by J. Ji and L. Wang in 2022. As the main result, an algorithm based on the QR factorization for completing a Mercedes-Benz frame containing specified elements is presented in this paper and two examples of Mercedes-Benz frames in R 4 are constructed to illustrate our approach. [ABSTRACT FROM AUTHOR]

In this paper, we study stochastic composite problems where the objective can be the composition of an outer single-valued function and an inner vector-valued mapping. In this stochastic composite optimization, the inner mapping can be expressed as an expectation over random component mappings. In this study, we propose two algorithms to address the generality and possible singularities of this problem and bound their sample complexities for finding an ϵ -stationary point. The first algorithm is the prox-linear hybrid stochastic gradient algorithm, which may achieve sample complexities of O (ϵ 2 τ − 5 / 2) and O (ϵ τ − 3 / 2) for the component mappings and their Jacobians respectively, where τ ∈ [ 0 , 1 ]. The second algorithm is the normalized proximal hybrid stochastic gradient algorithm, which takes advantage of the special structure of the regularizer. This algorithm may achieve sample complexities of O (ϵ 2 τ − 4) for both the component mappings and the Jacobians, where τ ∈ [ 5 / 4 , 7 / 4 ]. Numerical experiments prove that the two proposed algorithms are quite competitive with other existing algorithms. A real-life application in sparse portfolio selection problems is also promising. • We study a class of stochastic composite optimization problems. • We present two hybrid SGD algorithms for the considered problem. • We investigate the convergence rates and complexity of the algorithms. • We apply the proposed approach to the sparse portfolio selection problem. [ABSTRACT FROM AUTHOR]

In the paper the author presents a novel kind of the WZ factorization algorithm, namely a block WZ factorization algorithm. The aim of this new algorithm is to utilize the computational power of contemporary computers with hierarchical memory. In the paper, some properties of the matrix Z are given and analyzed. Next, a version of the block WZ factorization is presented. The author shows that such a block WZ factorization exists for strictly diagonally dominant matrices. The computational cost of this block algorithm is presented. The time and the accuracy of proposed block WZ factorization algorithm for random dense square diagonally dominant matrices are reported. The block algorithm turned out to be faster even up to 300 times than the original WZ factorization. [ABSTRACT FROM AUTHOR]

Powell (1984) and Dai (2003) constructed respectively a counterexample to show that the Polak–Ribière–Polyak (PRP) conjugate gradient algorithm fails to globally converge for nonconvex functions even when the exact line search technique is used, which implies similar failure of the weak Wolfe–Powell (WWP) inexact line search technique. Does another inexact line search technique exist that can ensure global convergence for nonconvex functions? This paper gives a positive answer to this question and proposes a modified WWP (MWWP) inexact line search technique. An algorithm is presented that uses the MWWP inexact line search technique; the next point x k + 1 generated by the PRP formula is accepted if a positive condition holds, and otherwise, x k + 1 is defined by a technique for projection onto a parabolic surface. The proposed PRP conjugate gradient algorithm is shown to possess global convergence under inexact line search for nonconvex functions. Numerical performance of the proposed algorithm is shown to be competitive with those of similar algorithms. • A convex parabolic surface for projection is defined and a projection PRP algorithm is proposed. • Any unsatisfactory iteration points are projected onto this convex parabolic surface to get good points. • The PRP conjugate gradient algorithm for general functions under inexact line search is obtained. • Competitive numerical results are reported to show the good performance of the new algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the convergence of several optimize-then-discretize and discretize-then-optimize algorithms, based on either a second-order or a fourth-order finite difference discretization, for solving elliptic PDE-constrained optimal control problems. To ensure the convergence of a discretize-then-optimize algorithm, one well-accepted criterion is to design the discretization scheme such that the resulting discretize-then-optimize algorithm commutes with the corresponding optimize-then-discretize algorithm. In other words, both algorithms should give rise to exactly the same discrete optimality system. However, such a restrictive criterion is not trivial to fulfill. By investigating a distributed control problem governed by an elliptic equation, we first show that enforcing such a stringent condition of commutative property is only sufficient but not necessary for achieving the desired convergence. We then introduce some suitable H 1 semi-norm penalty/regularization terms to recover the lost convergence due to the inconsistency caused by the loss of commutativity. Numerical experiments are carried out to verify our theoretical analysis and also validate the effectiveness of our proposed regularization techniques. [ABSTRACT FROM AUTHOR]

Registration refers to the useful process of aligning two similar but different intensity image functions in order to either track changes or combine information. Variational models are capable of finding transform maps containing large and non-uniform deformations between such a pair of images. Since finding a transform map is an inverse problem, as with all models, suitable regularisation is necessary to overcome the non-uniqueness of the problem. In the case of diffusion type models regularisation terms impose smoothness on the transformation by minimising the gradient of the flow field. The diffusion model also coincides with the basic model for optical flow frameworks of Horn and Schunck (1981). The biggest drawback with variational models is the large computational cost required to solve the highly non-linear system of PDEs; Chumchob and Chen (2011) developed a non-linear multigrid (NMG) method to address this cost problem. However, a closer look at the analysis of the NMG scheme highlighted omissions which affected the convergence of the NMG scheme. Moreover, the NMG method proposed by Chumchob and Chen did not impose any control of non-physical folding which invalidates a map. This paper has proposed several key ideas. First we re-evaluate the analysis of the NMG method to show how the omissions in Chumchob and Chen (2011) have a noticeable impact on the convergence of the NMG method. In addition, we also provide a way of estimating the convergence rate of a solver on the coarsest grid in order to estimate the number of iterations that will be required to obtain a solution with appropriate accuracy. Second we propose an extension to the Chumchob–Chen NMG method which controls any folding within the deformation. Experimental results on the proposed multigrid framework demonstrate improvements in convergence and the accuracy of registrations compared with previous methods. [ABSTRACT FROM AUTHOR]

In this paper an improved version of a verification algorithm for solving nonlinear systems of equations based on Krawczyk operator is presented. Compared with the original algorithm, the improved verification algorithm not only saves computing time, but also computes a narrower (or at least the same) inclusion of the solution to nonlinear systems of equations for certain classes of problems. Numerical results demonstrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The reduction of a matrix to an upper J -Hessenberg form is a crucial step in the S R -algorithm (which is a Q R -like algorithm), structure-preserving, for computing eigenvalues and vectors, for a class of structured matrices. This reduction may be handled via the algorithm JHESS or via the recent algorithm JHMSH and its variants. The main drawback of JHESS (or JHMSH) is that it may suffer from a fatal breakdown, causing a brutal stop of the computations and hence, the S R -algorithm does not run. JHESS may also encounter near-breakdowns, source of serious numerical instability. In this paper, we focus on these aspects. We first bring light on the necessary and sufficient condition for the existence of the S R -decomposition, which is intimately linked to J -Hessenberg reduction. Then we will derive a strategy for curing fatal breakdowns and also for treating near breakdowns. Hence, the J -Hessenberg form may be obtained. Numerical experiments are given, demonstrating the efficiency of our strategies to cure and treat breakdowns or near breakdowns. [ABSTRACT FROM AUTHOR]

Image demosaicking and denoising are the first two key steps of the color image production pipeline. The classical processing sequence has for a long time consisted of applying denoising first, and then demosaicking. Applying the operations in this order leads to oversmoothing and checkerboard effects. Yet, it was difficult to change this order, because once the image is demosaicked, the statistical properties of the noise are dramatically changed and hard to handle by traditional denoising models. In this paper, we address this problem by a hybrid machine learning method. We invert the traditional color filter array (CFA) processing pipeline by first demosaicking and then denoising. Our demosaicking algorithm, trained on noiseless images, combines a traditional method and a residual convolutional neural network (CNN). This first stage retains all known information, which is the key point to obtain faithful final results. The noisy demosaicked image is then passed through a second CNN restoring a noiseless full-color image. This pipeline order completely avoids checkerboard effects and restores fine image detail. Although CNNs can be trained to solve jointly demosaicking–denoising end-to-end, we find that this two-stage training performs better and is less prone to failure. It is shown experimentally to improve on the state of the art, both quantitatively and in terms of visual quality. [ABSTRACT FROM AUTHOR]

The rational fraction system is a special nonlinear system, the existence of the denominator polynomial leads to the difficulty of identifying rational fraction models. Inspired by the gradient search and the Newton method, the gradient-based iterative algorithm and Newton iterative algorithm are presented to estimate the parameters of rational fraction system models. Furthermore, in order to avoid a large amount of calculation and complex equations encountered in the process of solving partial derivatives, the model transformation-based gradient iterative algorithm and the model transformation-based Newton iterative algorithm are proposed for parameter identification. Two examples are carried out to show the effectiveness of the proposed algorithms. This paper focuses on solving the identification problem of rational fraction systems. [ABSTRACT FROM AUTHOR]

NURBS method is the standard mathematical method for describing the shapes of curves/surfaces/volumes, and it is extensively used in computer-aided design, computer-aided manufacturing, and computer graphics. The injectivity of NURBS volumes means that they do not have self-intersections. Since the injectivity of parameterizations depend on the signs of their Jacobian functions, and the Jacobian of a NURBS volume is determined by the determinant of its tangent vectors in three directions, we first propose a method to compute the bounding vectors of the tangent cones of NURBS volume in this paper. Then the sufficient condition for the injectivity of NURBS volume is proposed. A checking algorithm is also presented. Some examples are given to verify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

The paper presents two-scale numerical algorithms for stress–strain analysis of porous media featured by self-contact at pore level. The porosity is constituted as a periodic lattice generated by a representative cell consisting of elastic skeleton and a void pore. Unilateral frictionless contact is considered between opposing surfaces of the pore. For the homogenized model derived in our previous work, we justify incremental formulations and propose several variants of two-scale algorithms which commute iteratively solving of the micro- and the macro-level contact subproblems. A dual formulation which takes advantage of the assumed microstructure periodicity and a small deformation framework, is derived for the contact problems at the micro-level. This enables to apply the semi-smooth Newton method. For the global, macrolevel step two alternatives are tested; one relying on a frozen contact identified at the microlevel, the other based on a reduced contact associated with boundaries of contact sets. Numerical examples of 2D deforming structures are presented as a proof of the concept. • A new two-scale variational method for unilateral self-contact in porous media is derived. • The proposed incremental formulation uses a consistent homogenized elastic tensor. • The new macroscopic contact method couples micro- and macro-scopic contact problems. • A dual formulation is proposed to solve local contact microproblems efficiently. • The presented model and solution method are easily extensible to fluid-filled pores. [ABSTRACT FROM AUTHOR]

The hard thresholding technique plays a vital role in the development of algorithms for sparse signal recovery. By merging this technique and heavy-ball acceleration method which is a multi-step extension of the traditional gradient descent method, we propose the so-called heavy-ball-based hard thresholding (HBHT) and heavy-ball-based hard thresholding pursuit (HBHTP) algorithms for signal recovery. It turns out that the HBHT and HBHTP can successfully recover a k -sparse signal if the restricted isometry constant of the measurement matrix satisfies δ 3 k < 0. 618 and δ 3 k < 0. 577 , respectively. The guaranteed success of HBHT and HBHTP is also shown under the conditions δ 2 k < 0. 356 and δ 2 k < 0. 377 , respectively. Moreover, the finite convergence of HBHTP and stability of the two algorithms are also established in this paper. Simulations on random problem instances are performed to compare the performance of the proposed algorithms and several existing ones. Empirical results indicate that the HBHTP performs very comparably to a few existing algorithms and it takes less average time to achieve the signal recovery than these existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we present a Lorentzian- ℓ p (0 < p < 1) optimization for recovering sparse signals from measurements corrupted by two types of impulsive noise. A smoothing strategy for nonconvex ℓ p norm is employed and an efficient Lorentzian based iterative algorithm is developed for solving the resulting optimization problem. We describe the detailed process of the algorithm and establish the convergence of the proposed algorithm. Numerical results show that our proposed algorithm performs effectively in recovering sparse signals from impulsive measurement noise compared with the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In previous papers, the authors introduced and characterized a class of matrices called { K , s + 1 } -potent. Also, they established a method to construct these matrices. The purpose of this paper is to solve the associated inverse problem. Several algorithms are developed in order to find all involutory matrices K satisfying K A s + 1 K = A for a given matrix A ∈ C n × n and a given natural number s . The cases s = 0 and s ≥ 1 are separately studied since they produce different situations. In addition, some examples are presented showing the numerical performance of the methods. [ABSTRACT FROM AUTHOR]

Abstract The linearized alternating direction method of multipliers (ADMM), with indefinite proximal regularization, has been proved to be efficient for solving separable convex optimization subject to linear constraints. In this paper, we present a generalization of linearized ADMM (G-LADMM) to solve two-block separable convex minimization model, which linearizes all the subproblems by choosing a proper positive-definite or indefinite proximal term and updates the Lagrangian multiplier twice in different ways. Furthermore, the proposed G-LADMM can be expressed as a proximal point algorithm (PPA), and all the subproblems are just to estimate the proximity operator of the function in the objective. We specify the domain of the proximal parameter and stepsizes to guarantee that G-LADMM is globally convergent. It turns out that our convergence domain of the proximal parameter and stepsizes is significantly larger than other convergence domains in the literature. The numerical experiments illustrate the improvements of the proposed G-LADMM to solve LASSO and image decomposition problems. [ABSTRACT FROM AUTHOR]