Search

Gautschi proposed a method for computing incomplete beta functions I x (a , b) using Miller's algorithm with a three-term recurrence relation and showed a computation program in ALGOL. In this paper, first, Miller's algorithm using the recurrence relation satisfied by f k (x) = I x (a + k , b) is described. Next, another solution that is first-order independent of f k (x) of the recurrence relation is given, and its general solution can be expressed as a linear sum of these. Using this general solution, an error analysis for the function I x (a , b) is performed for the first time. The relative error of the function values is then expressed in a new formula to a trend of the error behavior. Also, Miller's algorithm with a normalizing sum is explained and its error analysis is performed. Since Miller's algorithm requires a predefined number of iterations of the recurrence relation, it is necessary to repeat the computation of the recurrence relation with increasing number of iterations until the required accuracy will be met. Therefore, in this paper, we apply Deuflhard's algorithm, which can automatically obtain the function value with the required accuracy. This algorithm requires far fewer iterations than Gautschi's algorithm to obtain the same accuracy. [ABSTRACT FROM AUTHOR]

In this paper, efficient algorithms will be studied for solving nonlinear saddle point problems from the elliptic optimal control problems. Based on the preconditioned inexact Uzawa (PIU) algorithmic framework, an improved preconditioned inexact Uzawa (I-PIU) algorithmic framework will first be constructed for solving a class of block-structured nonlinear systems, which possess a more general form of the nonlinear saddle point systems. When the I-PIU algorithmic framework is used for solving the block-structured nonlinear systems, a detailed theoretical analysis of the global convergence properties is proposed under the case that the nonlinear operator is single-valued. Besides, a special I-PIU method is established based on a new matrix splitting technique at the first iterative step of the I-PIU algorithmic framework, and then we establish the detailed theoretical results of the convergence. Finally, some numerical experiments are given to demonstrate the efficiency of the I-PIU method. [ABSTRACT FROM AUTHOR]

The classical parareal algorithm for time-periodic problems, solving a periodic-like coarse problem, called the periodic parareal algorithm with periodic coarse problem (PP-PC), usually converges slowly. In this paper, we present a new parallel-in-time algorithm for time-periodic problems based on the classical PP-PC algorithm and the Krylov subspace method. In this new algorithm, a new propagator derived by the Krylov subspace is chosen as the coarse propagator instead of the classical coarse propagator in the PP-PC algorithm. And because of the special characteristic of time-periodic problems, the Krylov subspace enhanced PP-PC algorithm needs to solve a periodic coarse problem on the coarse time grid on each iteration. We provide two different kinds of theoretical bounds under different assumptions for the proposed algorithm. Numerical results illustrate our analysis with two effective theoretical bounds for the heat equation, the wave equation, and the viscous Burgers equation, where we could also find that the new proposed algorithm converges faster than the classical PP-PC algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a two-metric variable scaled forward-backward algorithm is proposed to solve the nonconvex noncontinuous composite optimization problems with ℓ 0 regularization. We first explore using a class of locally Lipschitz scale folded concave functions to approach the ℓ 0 . Then, a convex half-absolute method is proposed to precisely approximate these nonconvex nonsmooth functions. A double iterative algorithm is considered to solve the convex-relaxed composite optimization problems, where the inner iterative subproblem is solved using a well-designed two-metric variable scaled forward-backward algorithm. Specifically, while the variable metric in the forward step is derived by quasi-Newton methods, in the backward implicit step, however, the scaled matrix is chosen naturally derived by the inherent nonconvex properties of the established folded concave regularization function. The convergence of the proposed algorithm is analyzed, and the effectiveness of the proposed approach is validated by extensive numerical experiments, including sparse signal recovery, breast cancer data classification, and image restoration. [ABSTRACT FROM AUTHOR]

In the present paper, the regularity and finite difference methods for Caputo-Hadamard fractional differential equations with initial value singularity are taken into consideration. To overcome the weak singularity and enhance convergence precision, a fitted scheme on nonuniform meshes is applied to such problems. Firstly, based on L log , 2 - 1 σ approximation, the temporal convergence accuracy of fitted scheme for the sub-diffusion equations is O (N - min { 2 r α , 2 }) , where N denotes the number of time steps, α is the fractional order and r is the mesh grading parameter. It is indicated that the performance of the fitted scheme is better than that of the standard L log , 2 - 1 σ scheme on (i) exponent meshes (i.e., r = 1 ) and (ii) graded meshes with the optimal choice of the mesh grading. Secondly, a second-order fitted scheme on exponent meshes for the diffusion-wave equations is obtained. Furthermore, for the sake of improving the computational efficiency and demonstrating the effectiveness of the decomposition of the solution, the fast algorithm and further decomposition of the solution for the sub-diffusion equations are investigated. Ultimately, some examples are presented to verify the availability of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient than other existing methods. Moreover, our algorithm is optimal in the sense of the least number of complex inversions. On the practice side, our algorithm outperforms existing algorithms on randomly generated matrices. We argue that this algorithm can be used to improve the practical utility of recursive Strassen-type algorithms by providing the fastest possible base case for the recursive decomposition process when applied to quaternion matrices. [ABSTRACT FROM AUTHOR]

In this paper, for solving a broad class of large-scale nonconvex and nonsmooth optimization problems, we propose a stochastic two-step inertial Bregman proximal alternating linearized minimization (STiBPALM) algorithm with variance-reduced stochastic gradient estimators. And we show that SAGA and SARAH are variance-reduced gradient estimators. Under expectation conditions with the Kurdyka–Łojasiewicz property and some suitable conditions on the parameters, we obtain that the sequence generated by the proposed algorithm converges to a critical point. And the general convergence rate is also provided. Numerical experiments on sparse nonnegative matrix factorization and blind image-deblurring are presented to demonstrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we develop a branch and bound algorithm for globally solving linear multiplicative programs (LMP). The problem LMP is firstly converted to an equivalent problem (EP2), and then a novel linear relaxation technique is constructed for (EP2) to get the lower bound of the optimum of LMP. Subsequently, the presented region pruning technique aims to eliminate as many areas as possible where the optimal solution does not exist. The convergence analysis of the algorithm is provided, and the number of worst-case iterations required is estimated to obtain an ε -optimal solution. Finally, the numerical experiments show the effectiveness of our presented algorithm. [ABSTRACT FROM AUTHOR]

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long-time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lions et al. in 2001, for multiscale Hamiltonian systems. The first method involves constructing a correction operator to improve a given inaccurate coarse solver through solving a Procrustes problem using data collected online along parareal trajectories. The second method involves constructing an efficient, high-fidelity solver by a neural network trained with offline generated data. For the second method, we address the issues of effective data generation and proper loss function design based on the Hamiltonian function. We show proof-of-concept by applying the proposed methods to a Fermi-Pasta-Ulam (FPU) problem. The numerical results demonstrate that the Procrustes parareal method is able to produce solutions that are more stable in energy compared to the standard parareal. The neural network solver can achieve comparable or better runtime performance compared to numerical solvers of similar accuracy. When combined with the standard parareal algorithm, the improved neural network solutions are slightly more stable in energy than the improved numerical coarse solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the computation of the Lerch transcendent by means of the Gauss-Laguerre formula. An a priori estimate of the quadrature error, that allows to compute the number of quadrature nodes necessary to achieve an arbitrary precision, is derived. Exploiting the properties of the Gauss-Laguerre rule and the error estimate, a truncated approach is also considered. The algorithm used and its Matlab implementation are reported. The numerical examples confirm the reliability of this approach. [ABSTRACT FROM AUTHOR]

The BFGS method is a common and effective method for solving unconstrained optimization problems in quasi-Newton algorithm. However, many scholars have proven that the algorithm may fail in some cases for nonconvex problems under Wolfe conditions. In this paper, an adaptive projection BFGS algorithm is proposed naturally which can solve nonconvex problems, and the following properties are shown in this algorithm: ➀ The generation of the step size α j satisfies the popular Wolfe conditions; ➁ a specific condition is proposed which has sufficient descent property, and if the current point satisfies this condition, the ordinary BFGS iteration process proceeds as usual; ➂ otherwise, the next iteration point x j + 1 is generated by the proposed adaptive projection method. This algorithm is globally convergent for nonconvex problems under the weak-Wolfe-Powell (WWP) conditions and has a superlinear convergence rate, which can be regarded as an extension of projection BFGS method proposed by Yuan et al. (J. Comput. Appl. Math. 327:274-294, 2018). Furthermore, the final numerical results and the application of the algorithm to the Muskingum model demonstrate the feasibility and competitiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a projection-based hybrid spectral gradient algorithm for nonlinear equations with convex constraints, which is based on a certain line search strategy. Convex combination technique is used to design a novel spectral parameter that is inspired by some classical spectral gradient methods. The search direction also meets the sufficient descent condition and trust region feature. The global convergence of the proposed algorithm has been established under reasonable assumptions. The results of the experiment demonstrate the proposed algorithm is both more promising and robust than some similar methods, and it is also capable of handling large-scale optimization problems. Furthermore, we apply it to problems involving sparse signal recovery and blurred image restoration. [ABSTRACT FROM AUTHOR]

This paper constructs an interval partition linearization algorithm for solving minimax linear fractional programming (MLFP) problem. In this algorithm, MLFP is converted and decomposed into a series of linear programs by dividing the outer 1-dimensional space of the equivalent problem (EP) into polynomially countable intervals. To improve the computational efficiency of the algorithm, two new acceleration techniques are introduced, in which the regions in outer space where the optimal solution of EP does not exist are largely deleted. In addition, the global convergence of the proposed algorithm is summarized, and its computational complexity is illustrated to reveal that it is a fully polynomial time approximation scheme. Finally, the numerical results demonstrate that the proposed algorithm is feasible and effective. [ABSTRACT FROM AUTHOR]

In this paper, we propose a constraint preserving algorithm for the smallest Z-eigenpair of the compact Laplacian tensor of an even-uniform hypergraph, where Householder transform is employed and a family of modified conjugate directions with sufficient descent is determined. Besides, we prove that there exists a positive step size in the new constraint preserving update scheme such that the Wolfe conditions hold. Based on these properties, we prove the convergence of the new algorithm. Furthermore, we apply our algorithm to the hypergraph partitioning and image segmentation, and numerical results are reported to illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Singular value decomposition plays a prominent role in the theoretical study and numerical calculation of a quaternion matrix in applied sciences. This paper, by means of a complex representation of a quaternion matrix, studies the algorithm for the singular value decomposition of a quaternion matrix, and derives a complex structure-preserving algorithm for the singular value decomposition of a quaternion matrix. This paper also gives two examples to demonstrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the application of the fading regularization method with the method of fundamental solutions (MFS) to the ill-posed Cauchy-Stokes problem. For both smooth and piecewise smooth two-dimensional geometries, we present a numerical reconstruction of the missing velocity and normal stress tensor on an inaccessible part of the boundary based on knowledge of over-prescribed noisy data acquired on the remaining accessible boundary part. Three numerical examples demonstrate the proposed numerical algorithm's accuracy, convergence, stability, and efficiency, as well as its ability to deblur the data. [ABSTRACT FROM AUTHOR]

This paper presents a new sufficiently descent algorithm for system of nonlinear equations where the underlying operator is pseudomonotone. The conditions imposed on the proposed algorithm to achieve convergence are Lipschitz continuity and pseudomonotonicity which is weaker than monotonicity assumption forced upon many algorithms in this area found in the literature. Numerical experiments on selected test problems taken from the literature validate the efficiency of the new algorithm. Moreover, the new algorithm demonstrates superior performance in comparison with some existing algorithms. Furthermore, the proposed algorithm is applied to reconstruct some disturbed signals. [ABSTRACT FROM AUTHOR]

In Meurant et al. (Numer. Algorithms 88(3), 1337–1359, 2021), we presented an adaptive estimate for the energy norm of the error in the conjugate gradient (CG) method. Here we consider algorithms for solving linear approximation problems with a general, possibly rectangular matrix that are based on applying CG to a system with a positive (semi-)definite matrix built from the original matrix. We discuss algorithms based on Hestenes–Stiefel-like implementation (often called CGLS and CGNE in the literature) as well as on bidiagonalization (LSQR and CRAIG), and both unpreconditioned and preconditioned variants. Each algorithm minimizes a certain quantity at each iteration (within the current Krylov subspace), that is related to some error norm. We call this "the quantity of interest". We show that the adaptive estimate used in CG can be extended for these algorithms to estimate the quantity of interest. Throughout, we emphasize the applicability of the estimate during computations in finite-precision arithmetic. We carefully derive the relations from which the estimate is constructed, without exploiting the global orthogonality that is not preserved during computation. We show that the resulting estimate preserves its key properties: it can be cheaply evaluated, and it is numerically reliable in finite-precision arithmetic under some mild assumptions. These properties make the estimate suitable for use in stopping the iterations. The numerical experiments confirm the robustness and satisfactory behaviour of the estimate. [ABSTRACT FROM AUTHOR]

We propose a variation of the forward–backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility to the algorithm and can be chosen arbitrarily as long as they together satisfy a norm condition. We present special cases where the deviation vectors, selected as predetermined linear combinations of previous iterates, always meet the norm condition. Notably, we introduce an algorithm employing a scalar parameter to interpolate between the conventional forward–backward splitting scheme and an accelerated O 1 n 2 -convergent forward–backward method that encompasses both the accelerated proximal point method and the Halpern iteration as special cases. The existing methods correspond to the two extremes of the allowed scalar parameter range. By choosing the interpolation scalar near the midpoint of the permissible range, our algorithm significantly outperforms these previously known methods when addressing a basic monotone inclusion problem stemming from minimax optimization. [ABSTRACT FROM AUTHOR]

In this paper, we design a low-cost feasible projection algorithm for variational inequalities by replacing the projection onto the feasible set with the projection onto a ball. In each iteration, it only needs to calculate the value of the mapping once, and the projection onto the ball contained in the feasible set (which has an explicit expression), so the algorithm is easier to implement and feasible. The convergence of the algorithm is proved when the Slater condition holds for the feasible set and the mapping is pseudomonotone, Lipschitz continuous. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

This paper proposes an accelerating outer space algorithm for globally solving generalized linear multiplicative problems (GLMP). Utilizing the logarithmic and exponential function properties, we begin with transforming the GLMP into an equivalent problem (EP) by introducing some outer space variables. Then, a two-stage outer space relaxation method is developed to convert the EP into a series of relaxed linear problems. Furthermore, to improve the convergence speed of the algorithm, we develop some accelerating techniques to remove the domains that do not contain a global optimal solution. Then, fusing the linear relaxed method and accelerating techniques into the branch-and-bound framework, we provide the accelerating outer space algorithm for solving the EP. Additionally, we analyze the global convergence of the proposed algorithm. Meanwhile, by investigating the algorithmic complexity, we estimate the maximum iterations required by the proposed algorithm in the worst case. Finally, in contrast to other algorithms in the currently known literature, numerical experimental results indicate that the proposed algorithm is feasible with better efficiency and robustness. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a descent algorithm for solving nonsmooth nonconvex optimization problems. The proposed method is based on the proximal bundle algorithm and the gradient sampling method and uses the advantages of both. In addition, this algorithm has the ability to handle inexact information, which creates additional challenges. The global convergence is proved with probability one. More precisely, every accumulation point of the sequence of serious iterates is either a stationary point if exact values of gradient are provided or an approximate stationary point if only inexact information of the function and gradient values is available. The performance of the proposed algorithm is demonstrated using some academic test problems. We further compare the new method with a general nonlinear solver and two other methods specifically designed for nonconvex nonsmooth optimization problems. [ABSTRACT FROM AUTHOR]

This paper considers a high-order numerical method for a computed solution of multidimensional convection-diffusion-reaction equation with time-fractional derivative subjected to appropriate initial and boundary conditions. The stability and error estimates of the proposed numerical approach are analyzed using the L ∞ (0 , T ; L 2) -norm. The theoretical study suggests that the new technique is unconditionally stable and temporal accurate with order O(τ2+α), where τ denotes the time step and 0 < α < 1. This result shows that the developed algorithm is faster and more efficient than a broad range of numerical techniques widely studied in the literature for the considered problem. Numerical experiments confirm the theory and they indicate that the proposed numerical scheme converges with accuracy O(τ2+α + h4), where h represents the space step. [ABSTRACT FROM AUTHOR]

Adopting a suitable approximation strategy can both enhance the robustness and improve the efficiency of the numerical algorithms. In this paper, we suggest combining two approximation criteria, the absolute one and the relative one, to the asymmetric forward-backward-adjoint splitting (AFBA) algorithm for a class of convex-concave saddle point problems, resulting in two inexact AFBA variants. These two approximation criteria are low-cost, since verifying them just involves the subgradient of a certain function. For both the absolute error AFBA and the relative error AFBA, we establish the global convergence and the O(1/N) convergence rate measured by the gap function in the ergodic sense, where N is the number of iterations. For the absolute error AFBA, we show that it possesses an O(1/N2) (linear convergence) rate of convergence, under the assumption that a part of (both) the underlying functions are strongly convex. We report some numerical results which demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a second-order partitioned method with multiple-time-step technique for the nonstationary Stokes-Darcy model. This method allows different time steps in different subdomains and improves the accuracy by the time filters. Besides, by designing new error estimate and time step adjustment strategy, we extend this method to variable timestep and develop single and double adaptive algorithms. Constant and variable time step tests are given to confirm the theoretical analysis and illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of the iteration number k, we establish global O (1 / k) pointwise and O (1 / k) ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case. Since the symmetric proximal ADMM can be seen as a class of ADMM variants, the new algorithm as well as its convergence rates generalize, in particular, many others in the literature. Numerical experiments illustrating the practical advantages of the method are reported. To the best of our knowledge, this work is the first one to study an inexact version of the symmetric proximal ADMM. [ABSTRACT FROM AUTHOR]

In this paper, we propose a filter sequential adaptive regularization algorithm using cubics (ARC) for solving nonlinear equality constrained optimization. Similar to sequential quadratic programming methods, an ARC subproblem with linearized constraints is considered to obtain a trial step in each iteration. Composite step methods and reduced Hessian methods are employed to tackle the linearized constraints. As a result, a trial step is decomposed into the sum of a normal step and a tangential step which is computed by a standard ARC subproblem. Then, the new iteration is determined by filter methods and ARC framework. The global convergence of the algorithm is proved under some reasonable assumptions. Preliminary numerical experiments and comparison results are reported. [ABSTRACT FROM AUTHOR]

When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z2 − c has the same form regardless of which algorithm has been used. In this paper, we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree d polynomials p(z) = zd − c. Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algorithms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the continuous relaxation reformulation of cardinality-constrained optimization problems that has become more popular in recent years and propose a new sequential optimality condition (approximate stationarity) for cardinality-constrained optimization problems, which is proved to be a genuine necessary optimality condition that does not require any constraint qualification to hold. We compare this condition with the rest of the sequential optimality conditions and prove that our condition is stronger and closer to the local minimizer. A problem-tailored regularity condition is proposed, and we show that this regularity condition ensures that the approximate stationary point proposed in this paper is the exact stationary point and is the weakest constraint qualification with this property. Finally, we apply the results of this paper to safeguarded augmented Lagrangian method and prove that the algorithm converges to the approximate stationary point proposed in this paper under mild assumptions, the existing theoretical results of this algorithm are further enhanced. [ABSTRACT FROM AUTHOR]

Distributed algorithms are receiving renewed attention across multiple disciplines due to the dramatically increasing demand of big data processing. We consider a class of consensus optimization problems over a static network system of multiple agents, where each of local cost functions is a sum of a smooth function and a non-Lipschitz regularization term. This kind of problems is widely found in scientific and engineering areas such as machine learning and data analysis. Inspired by Bian and Chen (SIAM J. Opt. 23(3), 1718–1741 2017), we propose a decentralized smoothing quadratic regularization algorithm (abbreviated as D-SQRA) for solving the composite consensus problem with non-Lipschitz singularities. To some extent, D-SQRA can be seen as an extension in the decentralized setup of the smoothing quadratic regularization algorithm (SQRA) proposed in (SIAM J. Opt. 23(3), 1718–1741 2017). Our main contribution is to show that D-SQRA can inherit the theoretical properties of its centralized counterpart, i.e., SQRA, in both sides of convergence and worst-case iteration complexity to achieve an ϵ scaled stationary point. We also present some numerical examples on sparse sensing problems based on synthetic and real datasets to corroborate the effectiveness of the proposed decentralized algorithm. [ABSTRACT FROM AUTHOR]

An optimization-based tool for flow predictions in natural rivers is introduced assuming that some physical characteristics of a river within a spatial-time domain [ x min , x max ] × [ t min , t today ] are known. In particular, it is assumed that the bed elevation and width of the river are known at a finite number of stations in [ x min , x max ] and that the flow-rate at x = x min is known for a finite number of time instants in [ t min , t today ] . Using these data, given t future > t today and a forecast of the flow-rate at x = x min and t = t future , a regression-based algorithm informed by partial differential equations produces predictions for all state variables (water elevation, depth, transversal wetted area, and flow-rate) for all x ∈ [ x min , x max ] and t = t future . The algorithm proceeds by solving a constrained optimization problem that takes into account the available data and the fulfillment of Saint-Venant equations for one-dimensional channels. The effectiveness of this approach is corroborated with flow predictions of a natural river. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm for finding a zero of a pseudomonotone vector field with a convex constraint on a Hadamard manifold. This new method is the combination of the hyperplane projection method with specially constructed search directions. The global convergence property of this algorithm is established under the assumptions that the constructed halfspace is closed and convex, the tangent vector field is continuous, and the solution set is nonempty. Numerical experiments show the efficiency of this new derivative-free iterative method. [ABSTRACT FROM AUTHOR]

As a generalization of k-tridiagonal matrices, many variations of (p,q)-tridiagonal matrices have attracted much attention over the years. In this paper, we present an efficient algorithm for numerically computing the inverses of n-square (p,q)-tridiagonal matrices under a certain condition. The algorithm is based on the combination of a bidiagonalization approach which preserves the banded structure and sparsity of the original matrix, and a recursive algorithm for inverting general lower bidiagonal matrices. Some numerical results with simulations in MATLAB implementation are provided in order to illustrate the accuracy and efficiency of the proposed algorithms, and its competitiveness with MATLAB built-in function. [ABSTRACT FROM AUTHOR]

The mutual coherence of a matrix, defined as the maximum absolute value of the normalized inner-products between different columns, is an important property that characterizes the similarity between different matrix columns. Redundant matrices with very low mutual coherence are referred to as incoherent redundant matrices which play an important role in mathematical signal processing tasks. The problem of minimizing the mutual coherence in a give matrix space where each matrix has normalized columns is called the coherence optimization problem. In this paper, we transform equivalently the coherence optimization problem as a rank constrained semidefinite ℓ ∞ -minimization problem. It is critical to analyze the projection operator onto the nonconvex set in the new matrix optimization constraints, i.e., the nonconvex set of symmetric positive semidefinite matrices whose rank is not greater than a give positive integer. By exploiting the projection operator, we express the nonconvex set mentioned above as the set of zero roots of a Difference of two Convex (DC) functions. For the convex function related to the projection operator in the DC function, we characterize its properties and obtain the concise form of its subdifferential. With the help of the DC function, a new algorithm based on DCA (DC Algorithms) is proposed to solve the Coherence Optimization problem, and thus the proposed algorithm is called DCACO for short. We also study the convergence analysis of DCACO. An advantage of DCACO is that subproblems in each iteration have closed-form solutions. Experimental results demonstrate that DCACO leads to state-of-art performance on generating highly incoherent redundant matrices, and DCACO can also compete with several other algorithms on designing optimized projection matrices for improving the performance of compressed sensing. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem of nonlinear systems with mixed control variables and probability constraints. To obtain a numerical solution of this optimal control problem, our target is to formulate this problem as a constrained nonlinear parameter optimization problem (CNPOP), which can be solved by using any gradient-based numerical computation method. Firstly, some binary functions are introduced for each value of the discrete-valued control variable (DCV). Following that, we relax these binary functions and impose a penalty term on the relaxation such that the solution of the resulting relaxed problem (RP) can converge to the solution of the original problem as the penalty parameter increases. Secondly, we introduce a simple initial transformation for the probability constraints. Following that, an adaptive sample approximation method (ASAM) and a novel smooth approximation technique (NSAT) are adopted to formulate the probability constraints as some deterministic constraints. Thirdly, a control parameterization approach (CPA) is used to transform the deterministic problem (i.e., an infinite dimensional problem) into a finite dimensional CNPOP. Fourthly, in order to combine the advantages of limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithms and differential evolution (DE) algorithms, a L-BFGS-based DE (L-BFGS-DE) algorithm is proposed for solving the resulting approximation problem based on an improvied L-BFGS (IL-BFGS) method and an improved DE (IDE) algorithm. Following that, we establish the convergence result of the L-BFGS-DE algorithm. The L-BFGS-DE algorithm consists of two stages. The objectives of the first and second stages are to obtain a probable position of the global solution and to accelerate the convergence rate, respectively. In the IL-BFGS method, we propose a novel updating rule (NUR), which uses not only the gradient information of the objective function but also the value of the objective function. This will improved the performance of the IL-BFGS method. In the IDE algorithm, a novel adaptive parameter adjustment (NAPA) method, a novel population size decrease (NPSD) strategy, and an improved mutation (IM) scheme are proposed to improve its performance. Finally, an anti-cancer drug therapy problem (ADTP) is further extended to illustrate the effectiveness of the L-BFGS-DE algorithm by taking into account some probability constraints. Numerical results show that the L-BFGS-DE algorithm has good performance and can obtain a stable and robust performance when considering the small noise perturbations in initial state. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of hybrid inertial and contraction proximal point algorithm for the variational inclusion problem of the sum of two mappings in Hilbert spaces. We prove that the proposed algorithm converges strongly to a solution of the variational inclusion problem whenever its solution set is nonempty and the single-valued mapping f is Lipschitz continuous, monotone, and the set-valued mapping A is maximal monotone in infinite-dimensional real Hilbert spaces. Our work generalize and extend some related existing results in the literature. Finally, we illustrate the numerical performance of our Algorithm 1 and we give an application to the split feasibility problem. [ABSTRACT FROM AUTHOR]

We formulate the Multi-Constrained Dynamic Traffic Assignment (DTA) problem as an instance of the nonlinear composite problem. To solve the problem, this paper introduces then the penalized nonlinear alternating direction method of multipliers (ADMM), a numerical algorithm that combines the nonlinear ADMM algorithm with the external penalty method. Numerical results are then presented, analyzed and compared against those obtained by applying the Reformulation-Linearization Technique (RLT)-based convex relaxation method together with piecewise linear approximation of the objective function. [ABSTRACT FROM AUTHOR]

This paper proposes a new definition for the term filled function, which weakens the third condition of the filling properties. Based on this new definition, a general form for parameter-free filled functions is created. One of the specific parameter-free filled functions of the proposed general form is then implemented in some comparable global optimization problems. The data generated from the numerical process reveals that the proposed filled functions are dependable and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we study the split feasibility problem with multiple output sets in Hilbert spaces. For solving the aforementioned problem, we propose two new self-adaptive relaxed CQ algorithms which involve computing of projections onto half-spaces instead of computing onto the closed convex sets, and it does not require calculating the operator norm. We establish a weak and a strong convergence theorems for the proposed algorithms. We apply the new results to solve some other problems. Finally, we present some numerical examples to show the efficiency and accuracy of our algorithm compared to some existing results. Our results extend and improve some existing methods in the literature. [ABSTRACT FROM AUTHOR]

In a very interesting paper (SIAM J. Control Optim. 37(3): 765–776, 1999), Solodov and Svaiter introduced an effective projection algorithm with linesearch for finding a solution of a variational inequality problem (VIP) in Euclidean space. They showed that the iterative sequence generated by their algorithm converges to a solution of (VIP) under the main assumption that the cost mapping is pseudomonotone and continuous. In this paper, we propose to modify this algorithm for solving variational inequality problems in which the cost mapping is not required to be satisfied any pseudomonotonicity. Moreover, we do not use the embedded projection methods as in methods used in literature and the linesearch procedure is not necessary when the cost mapping is Lipschitz. Several numerical examples are also provided to illustrate the efficient of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic optimization problem over the fixed point sets of quasinonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian hierarchical optimization problems over complicated sets, such as the intersection of many closed convex sets, the set of all minimizers of a nonsmooth convex function, and the intersection of sublevel sets of nonsmooth convex functions. We focus on adaptive learning rate optimization algorithms, which adapt step-sizes (referred to as learning rates in the machine learning field) to find optimal solutions quickly. We then propose a Riemannian stochastic fixed point optimization algorithm, which combines fixed point approximation methods on Riemannian manifolds with the adaptive learning rate optimization algorithms. We also give convergence analyses of the proposed algorithm for nonsmooth convex and smooth nonconvex optimization. The analysis results indicate that, with small constant step-sizes, the proposed algorithm approximates a solution to the problem. Consideration of the case in which step-size sequences are diminishing demonstrates that the proposed algorithm solves the problem with a guaranteed convergence rate. This paper also provides numerical comparisons that demonstrate the effectiveness of the proposed algorithms with formulas based on the adaptive learning rate optimization algorithms, such as Adam and AMSGrad. [ABSTRACT FROM AUTHOR]

We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary and sufficient conditions under which a convex combination and composition of strict pseudocontractions as well as demicontractions that share a common fixed point is again a strict pseudocontraction or a demicontraction, respectively. Moreover, we introduce a generalized relaxation of composition of demicontraction and give its properties. We" apply these properties to prove the weak convergence of a class of algorithms that is wider than the Douglas–Rachford algorithm and projected Landweber algorithms. We have also presented two numerical examples, where we compare the behavior of the presented methods with the Douglas–Rachford method. [ABSTRACT FROM AUTHOR]

We introduce two new self-adaptive algorithms for solving the split feasibility problem with multiple output sets in Hilbert space. Our algorithms do not use the least squares method, and their step sizes do not depend on the norms of the transfer mappings. Furthermore, two relaxed iterative algorithms are introduced when the feasibility sets are sublevel sets of convex functions. [ABSTRACT FROM AUTHOR]

We introduce four novel relaxed CQ algorithms with alternating inertial for solving split feasibility problems in real Hilbert spaces. The proposed algorithms employ a new non-monotonic adaptive step size criterion and utilize two different step sizes in each iteration. The weak convergence of the iterative sequences generated by the proposed algorithms is established under some weak conditions. Moreover, the Fejér monotonicity of the even subsequence with respect to the solution set is recovered. Two applications in signal denoising and image deblurring are given to illustrate the computational efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a two-sided, parallel Kogbetliantz-type algorithm for the hyperbolic singular value decomposition (HSVD) of real and complex square matrices is developed, with a single assumption that the input matrix, of order n, admits such a decomposition into the product of a unitary, a non-negative diagonal, and a J-unitary matrix, where J is a given diagonal matrix of positive and negative signs. When J = ±I, the proposed algorithm computes the ordinary SVD. The paper's most important contribution—a derivation of formulas for the HSVD of 2 × 2 matrices—is presented first, followed by the details of their implementation in floating-point arithmetic. Next, the effects of the hyperbolic transformations on the columns of the iteration matrix are discussed. These effects then guide a redesign of the dynamic pivot ordering, being already a well-established pivot strategy for the ordinary Kogbetliantz algorithm, for the general, n × n HSVD. A heuristic but sound convergence criterion is then proposed, which contributes to high accuracy demonstrated in the numerical testing results. Such a J-Kogbetliantz algorithm as presented here is intrinsically slow, but is nevertheless usable for matrices of small orders. [ABSTRACT FROM AUTHOR]

Recently, a number of efficient iteration methods for the solution of a special class of block 3 × 3 saddle point systems have been proposed by some authors. In order to easily evaluate the strong stability of these numerical algorithms and provide a practical and reliable termination criterion, in this paper, we perform the structured backward error analysis for this type of block 3 × 3 saddle point system and present an explicit and computable formula of the normwise structured backward error. Some numerical experiments are performed to demonstrate that our results can be used to easily test the stability of running algorithms, and the new stopping criterion based on the derived structured backward error is more suitable and efficient than the commonly used residue-based one. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized discrete Lotka-Volterra equation and explore its connections with symmetric orthogonal polynomials, Hankel determinants and convergence acceleration algorithms. Firstly, we extend the fully discrete Lotka-Volterra equation to a generalized one with a sequence of given constants { u 0 (n) } and derive its solution in terms of Hankel determinants. Then, it is shown that the discrete equation of motion is transformed into a discrete Riccati system for a discrete Stieltjes function, hence leading to a complete linearization. Besides, we obtain its Lax pair in terms of symmetric orthogonal polynomials by generalizing the Christoffel transformation for the symmetric orthogonal polynomials. Moreover, a generalization of the famous Wynn's ε-algorithm is also derived via a Miura transformation to the generalized discrete Lotka-Volterra equation. Finally, the numerical effects of this generalized ε-algorithm are discussed by applying to some linearly, logarithmically convergent sequences and some divergent series. [ABSTRACT FROM AUTHOR]

Theory of quasi-orthogonal polynomials is significantly related to constructing vector Padé approximations. The present paper introduces an efficient procedure to compute adjacent families of quasi-orthogonal polynomials as defined in Sadaka (Appl. Numer. Math. 21:57–70, 1996) and Sadaka (Appl. Numer. Math. 24:483–499, 1997). The derived algorithm uses short recursive relations whose coefficients are written in terms of lower triangular block determinants. The strategy of computing such determinants is given and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the computational problem of the quaternion equality constrained least squares problem. First, applying the quaternion QR decomposition and an equivalence problem of the quaternion equality constrained least squares problem, we obtain the expressions of the general solution and the minimal norm solution of the quaternion equality constrained least squares problem. And then by using the real representation matrices of quaternion matrices, the special structure of the real representation matrices and the real structure-preserving algorithm of the quaternion QR decomposition, we purpose a real structure-preserving algorithm for the minimal norm solution of the quaternion equality constrained least squares problem. The purposed algorithm only involves real algebraic operations, and the number of real floating-point operations and assignments has been minimized. Finally, we give two examples, which show that our purposed algorithm is more efficient and time-saving than direct computation in the quaternion Toolbox for Matlab. [ABSTRACT FROM AUTHOR]

In this paper, based on the Gauss transformation of a quaternion matrix, we study the full rank decomposition of a quaternion matrix, and obtain a direct algorithm and complex structure-preserving algorithm for full rank decomposition of a quaternion matrix. In addition, we expand the application of the above two full rank decomposition algorithms and give a fast algorithm to calculate the quaternion linear equations. The numerical examples show that the complex structure-preserving algorithm is more efficient. Finally, we apply the structure-preserving algorithm of the full rank decomposition to the sparse representation classification of color images, and the classification effect is well. [ABSTRACT FROM AUTHOR]