一种新型压缩预处理CGS算法

CGS算法是求解大型非对称线性方程组的常用算法，然而该算法无极小残差性质，因此它常因出现较大的中间剩余向量而出现典型的不规则收敛行为．本根据IRA方法提出了一种压缩预处理CGS方法，数值实验表明这种算法在一定程度上减小了迭代算法在收敛过程中的剩余问题，从而使得算法具有更好的稳定性，该法构造简单，减少了收敛次数，加快了收敛速度．     

A New Gradient Method for Ill-Posed Problems

In this paper, we present a new gradient method for linear and nonlinear ill-posed problems F(x) = y. Combined with the discrepancy principle as stopping rule it is a regularization method that yields convergence to an exact solution if the operator F satisfies a tangential cone condition. If the exact solution satisfies smoothness conditions, then even convergence rates can be proven. Numerical results show that the new method in most cases needs less iteration steps than Landweber iteration, the steepest descent or minimal error method.     

A heuristic algorithm for nonlinear programming

In this paper, a heuristic algorithm for nonlinear programming is presented. The algorithm uses two search directions, and the Hessian of the Lagrangian function is approximated with the BFGS secant update. We show that the sequence of iterates convergeq-superlinearly if the sequence of approximating matrices satisfies a particular condition. Numerical results are presented.     

MULTIGRID METHODS FOR THE GENERALIZED STOKES EQUATIONS BASED ON MIXED FINITE ELEMENT METHODS

1. IntroductionIn this paper, we consider the fOllowing generalized stationary Stokes equations:where fl is a bounded convex domain in R', u represents the velocity of fluid, p its pressure; Fand G are external fOrce and source terms. Note that the source…     

On solenoidal high‐degree polynomial approximations to solutions of the stationary Stokes equations

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the velocity and pressure functions satisfying the Stokes equations. Error estimates show convergence of the method. An implementation using polynomial bases is described that permits the use of the continuous approximations of the h‐p finite element method and exactly satisfies the solenoidal requirement. We express the error estimates in terms of the diameter h of a cell and degree p of the approximation on each cell. Examples of 10^th degree polynomial approximations are described that substantiate the theoretical estimates. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 480–493, 2000     

Preconditioned GMRES methods with incomplete Givens orthogonalization method for large sparse least-squares problems

We propose to precondition the GMRES method by using the incomplete Givens orthogonalization (IGO) method for the solution of large sparse linear least-squares problems. Theoretical analysis shows that the preconditioner satisfies the sufficient condition that can guarantee that the preconditioned GMRES method will never break down and always give the least-squares solution of the original problem. Numerical experiments further confirm that the new preconditioner is efficient. We also find that the IGO preconditioned BA-GMRES method is superior to the corresponding CGLS method for ill-conditioned and singular least-squares problems.     

On approximating the solution of the non-stationary Stokes equations using the cell discretization algorithm

The cell discretization algorithm, a nonconforming extension of the finite element method, is used to obtain approximations to the velocity and pressure satisfying the nonstationary Stokes equations. Error estimates show convergence of the approximations. An implementation using polynomial bases is described that permits the use of the continuous approximations of the h–p finite element method and exactly satisfies the solenoidal requirement. We express the error estimates in terms of the diameter h of a cell and the degree p of the approximation on each cell. Results of an experiment with p10 are presented that confirm the theoretical estimates.     

A ROBUST SQP METHOD FOR OPTIMIZATION WITH INEQUALITY CONSTRAINTS

A new algorithm for inequality constrained optimization is presented, which solves a linear programming subproblem and a quadratic subproblem at each iteration. The algorithm can circumvent the difficulties associated with the possible inconsistency of QP subproblem of the original SQP method. Moreover, the algorithm can converge to a point which satisfies a certain first-order necessary condition even if the original problem is itself infeasible. Under certain condition, some global convergence results are proved and local superlinear convergence results are also obtained. Preliminary numerical results are reported.     

Convergence to a second-order point of a trust-region algorithm with a nonmonotonic penalty parameter for constrained optimization

In a recent paper (Ref. 1), the author proposed a trust-region algorithm for solving the problem of minimizing a nonlinear function subject to a set of equality constraints. The main feature of the algorithm is that the penalty parameter in the merit function can be decreased whenever it is warranted. He studied the behavior of the penalty parameter and proved several global and local convergence results. One of these results is that there exists a subsequence of the iterates generated by the algorithm that converges to a point that satisfies the first-order necessary conditions.In the current paper, we show that, for this algorithm, there exists a subsequence of iterates that converges to a point that satisfies both the first-order and the second-order necessary conditions.This research was supported by the Rice University Center for Research on Parallel Computation, Grant R31853, and the REDI Foundation.     

一类拟牛顿非单调信赖域算法及其收敛性

本文提出了一类求解无约束最优化问题的非单调信赖域算法.将非单调Wolfe线搜索技术与信赖域算法相结合,使得新算-法不仅不需重解子问题,而且在每步迭代都满足拟牛顿方程同时保证目标函数的近似Hasse阵Bk的正定性.在适当的条件下,证明了此算法的全局收敛性.数值结果表明该算法的有效性.     

推广AS-GN混合共轭梯度算法

本文提出了一种求解无约束优化问题的新算法,使Touati-Ahmed, Storey提出的混合共轭梯度法(以下简称AS)和Gilbert, Nocedal提出的混合共轭梯度法(以下简称GN)成为新算法在精确线性搜索下的特例.通过构造新的$\beta_{k}$计算公式,新算法自然满足下降性条件,且这个性质与线性搜索和目标函数的凸性均无关.在一般的条件下,我们证明了新算法的全局收敛性.数值结果表明该算法对测试函数是有效的.     

梯度Q-线性收敛的光滑凸极小化的一阶算法

受性能估计问题（PEP）方法的启发,通过考察最坏函数误差的收敛边界（即效率）,优化了迭代点对应的梯度满足Q-线性收敛的光滑凸极小化的一阶方法的步长系数。介绍新的有效的一阶方法,称为QGM,具有与优化梯度法（OGM）类似的计算有效形式。     

On quasi-Newton and pseudo-Newton algorithms

This paper surveys some of the existing approaches to quasi-Newton methods and introduces a new way for constructing inverse Hessian approximations for such algorithms. This new approach is based on restricting Newton's method to subspaces over which the inverse Hessian is assumed to be known, while expanding this subspace using gradient information. It is shown that this approach can lead to some well-known formulas for updating the inverse Hessian approximation. Deriving such updates through this approach provides new understanding of these formulas and their relation to the pseudo-Newton-Raphson algorithm.     

AN INTERIOR TRUST REGION ALGORITHM FOR NONLINEAR MINIMIZATION WITH LINEAR CONSTRAINTS

AbstractAn interior trust-region-based algorithm for linearly constrained minimization problems is proposed and analyzed. This algorithm is similar to trust region algorithms for unconstrained minimization: a trust region subproblem on a subspace is solved in each iteration. We establish that the proposed algorithm has convergence properties analogous to those of the trust region algorithms for unconstrained minimization. Namely, every limit point of the generated sequence satisfies the Krush-Kuhn-Tucker (KKT) conditions and at least one limit point satisfies second order necessary optimality conditions. In addition, if one limit point is a strong local minimizer and the Hessian is Lipschitz continuous in a neighborhood of that point, then the generated sequence converges globally to that point in the rate of at least 2-step quadratic. We are mainly concerned with the theoretical properties of the algorithm in this paper. Implementation issues and adaptation to large-scale problems will be addressed in a     

An ODE-Based Trust Region Filter Algorithm for Unconstrained Optimization

In this article, an ODE-based trust region filter algorithm for unconstrained optimization is proposed. It can be regarded as a combination of trust region and filter techniques with ODE-based methods. Unlike the existing trust-region-filter methods and ODE-based methods, a distinct feature of this method is that at each iteration, a reduced linear system is solved to obtain a trial step, thus avoiding solving a trust region subproblem. Under some standard assumptions, it is proven that the algorithm is globally convergent. Preliminary numerical results show that the new algorithm is efficient for large scale problems.     

A derivative-free algorithm for systems of nonlinear inequalities

Recently a new derivative-free algorithm has been proposed for the solution of linearly constrained finite minimax problems. This derivative-free algorithm is based on a smoothing technique that allows one to take into account the non-smoothness of the max function. In this paper, we investigate, both from a theoretical and computational point of view, the behavior of the minmax algorithm when used to solve systems of nonlinear inequalities when derivatives are unavailable. In particular, we show an interesting property of the algorithm, namely, under some mild conditions regarding the regularity of the functions defining the system, it is possible to prove that the algorithm locates a solution of the problem after a finite number of iterations. Furthermore, under a weaker regularity condition, it is possible to show that an accumulation point of the sequence generated by the algorithm exists which is a solution of the system. Moreover, we carried out numerical experimentation and comparison of the method against a standard pattern search minimization method. The obtained results confirm that the good theoretical properties of the method correspond to interesting numerical performance. Moreover, the algorithm compares favorably with a standard derivative-free method, and this seems to indicate that extending the smoothing technique to pattern search algorithms can be beneficial.     

解一类结构变分不等式问题的非精确并行交替方向法

带线性约束的具有两分块结构的单调变分不等式问题, 出现在许多现代应用中, 如交通和经济问题等. 基于该问题良好的可分结构, 分裂型算法被广泛研究用于其求解. 提出新的带回代的非精确并行交替方向法解该类问题, 在每一步迭代中,首先以并行模式通过投影得到预测点, 然后对其校正得到下一步的迭代点. 在压缩型算法的理论框架下, 在适当条件下证明了所提算法的全局收敛性. 数值结果表明了算法的有效性. 此外, 该算法可推广到求解具有多分块结构的问题.     

一类带非单调线搜索的信赖域算法

通过将非单调Wolfe线搜索技术与传统的信赖域算法相结合,我们提出了一类新的求解无约束最优化问题的信赖域算法.新算法在每一迭代步只需求解一次信赖域子问题,而且在每一迭代步Hesse阵的近似都满足拟牛顿条件并保持正定传递.在一定条件下,证明了算法的全局收敛性和强收敛性.数值试验表明新算法继承了非单调技术的优点,对于求解某...     

A derivative-free three-term projection algorithm involving spectral quotient for solving nonlinear monotone equations

ABSTRACT

In this paper, we develop a three-term conjugate gradient method involving spectral quotient, which always satisfies the famous Dai-Liao conjugacy condition and quasi-Newton secant equation, independently of any line search. This new three-term conjugate gradient method can be regarded as a variant of the memoryless Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method with regard to spectral quotient. By combining this method with the projection technique proposed by Solodov and Svaiter in 1998, we establish a derivative-free three-term projection algorithm for dealing with large-scale nonlinear monotone system of equations. We prove the global convergence of the algorithm and obtain the R-linear convergence rate under some mild conditions. Numerical results show that our projection algorithm is effective and robust, and is more competitive with the TTDFP algorithm proposed Liu and Li [A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo. 2016;53:427–450].     

Linear Convergence of Iterative Soft-Thresholding

In this article a unified approach to iterative soft-thresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the so-called finite basis injectivity property or the minimizer possesses a so-called strict sparsity pattern. Moreover it is shown that the constants can be calculated explicitly in special cases (i.e. for compact operators). Furthermore, the techniques also can be used to establish linear convergence for related methods such as the iterative thresholding algorithm for joint sparsity and the accelerated gradient projection method.