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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, 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]

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]

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]

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]

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]

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]

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]

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 this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied to binary and ternary 4-point interpolatory subdivision schemes, we find that the corresponding coefficient matrices are strictly diagonally dominant, and so the evaluation processes are robust. For any individual irrational parameters between 0 and 1, its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound. For surface schemes generalizing univariate subdivision schemes with polynomial reproduction property, exact evaluation methods can also be derived by combining Stam’s method with that of this paper. [ABSTRACT FROM AUTHOR]

This paper introduces a hybrid algorithm of Nelder–Mead and scatter search algorithms called SSNM. Numerical results for some known test problems with varying dimensions from 2 to 100 variables prove its great performance for a bounded or constrained problem. This paper also has a considerable contribution to the joint economic lot sizing (JELS) problem literature. This problem is considered for a two-stage supply chain with price-sensitive demand for geometric, geometric-then-equal size and optimal shipment policies. Solution procedures are also developed which use the SSNM algorithm in their steps. These models and solution procedures are novel in the JELS literature. [ABSTRACT FROM AUTHOR]

To find new solutions through the information provided by previously found solution(s), this paper is first to develop a new singular transform to change the local basin-barrier structure of the original problem in an infinite-dimensional space and then to propose an augmented partial Newton method to solve the transformed problem on the solution set. Mathematical justifications of the new formulation and method are established. Details on algorithm implementation are presented. Examples for two very different variational PDE problems are carried out to illustrate the method. Their numerical results are displayed by profile–contour plots and error/convergence data. The formulation and method developed in this paper can also be applied to solve multiple fixed point problems. [ABSTRACT FROM AUTHOR]

This paper presents an accurate implicit re-initialization approach in the framework of a level set method. The improved method includes two schemes for keeping the zero level set unperturbed. The first scheme is to derive and use a new formula for the smoothing parameter in the conventional re-initialization equation based on the principle of the interface not moving. The second scheme is to reduce the local time step to avoid the interface moving across grid points when the sign of the level set function near the interface is changed. The example presented suggests that the new algorithm has a better approximation to the signed distance function and obtains a more accurate interface than the algorithm presented by Sussman et al. (1994). For shape and topology optimization of fluid, the effectiveness of the developed method in this paper was demonstrated by the internal and external flow examples. [ABSTRACT FROM AUTHOR]

In this paper new algorithms are being proposed for evaluation of highly oscillatory integrals (HOIs) with stationary point(s). The algorithms are based on modified Levin quadrature (MLQ) with multiquadric radial basis functions (RBFs) coupled with quadrature rules based on hybrid functions of order 8 (HFQ8) and Haar wavelets quadrature (HWQ) (Aziz et al. 2011). Part of the new procedure presented in this paper is comprised of transplanting monomials (which are used in the conventional Levin method) by the RBFs. The linear and Hermite polynomials based quadratures (Xiang, 2007) are being replaced by the new methods based on HWQ and HFQ8 respectively. Both the methods are merged with MLQ to obtain the numerical solution of highly oscillatory integrals having stationary points. The accuracy of the new methods is neither dampened by presence of the stationary point(s) nor by the large value of frequency parameter ω . Theoretical facts about the error analysis of the new methods are analyzed and proved. Numerical examples are included to show efficiency and accuracy of the new methods. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of finding poles of rational functions from function values on open curves in the complex plane. For this problem, Nara and Ando recently proposed an algorithm that reduces the problem to a system of linear equations through contour integration. The main aim of this paper is to analyze and improve this algorithm by giving a new interpretation to the algorithm in terms of orthogonal polynomials. It is demonstrated that the system of linear equations is not always uniquely solvable and that this difficulty can be remedied by doubling the number of the linear equations. Moreover, to cope with discretization errors caused by numerical integration, we introduce new polynomials similar, in spirit, to discrete orthogonal polynomials, which yield an algorithm free from discretization errors. [ABSTRACT FROM AUTHOR]

In this paper, using the properties of graph convergence of H(.,.)-accretive operators, we construct a perturbed algorithm for solving systems of ill-posed variational inclusions involving H(.,.)-accretive operators. Under some suitable conditions, we prove that the sequence generated by the proposed method is strongly convergent to the unique solution of the problem. The main results presented in this paper generalize and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents two different versions of an optimal control method for enforcing mass conservation in level set algorithms. The proposed PDE-constrained optimization procedure corrects a numerical solution to the level set transport equation so as to satisfy a conservation law for the corresponding Heaviside function. In the original version of this method, conservation errors are corrected by adding the gradient of a scalar control variable to the convective flux in the state equation. In the present paper, we investigate the use of vector controls. The alternative formulation offers additional flexibility and requires less regularity than the original method. The nonlinear system of first-order optimality conditions is solved using a standard fixed-point iteration. The new methodology is evaluated numerically and compared to the scalar control approach. [Copyright &y& Elsevier]

Abstract: Theoretical analyses of algorithms are important to understand their search behaviors and develop more efficient algorithms. Compared with the plethora of works concerning the empirical study of the differential evolution (DE), little theoretical research has been done to investigate the convergence properties of DE so far. This paper focuses on theoretical researches on the convergence of DE and presents a convergent DE algorithm. First of all, it is proved that the classical DE cannot converge to the global optimal set with probability 1 by using the property that it cannot escape from a local optimal set. Inspired by the characteristics of the elitist genetic algorithm, this paper proposed a modified DE to overcome the disadvantage. The proposed algorithm employs two operators that assist it in escaping from a local optimal set and enhance the diversity of the population. And it is then verified that the proposed algorithm is capable of converging to global optima with probability 1. The theoretical research of this paper is undertaken in a finite discrete set, and the analysis tool used is the Markov chain. The numerical experiments are conducted on a deceptive function and a set of benchmark functions. The experimental results support the theoretical analyses on the convergence performances of the classical and modified DE algorithm. [Copyright &y& Elsevier]

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]