On Newton-Like Methods for Solving Nonlinear Equations

In this paper, we present a family of general New to n-like methods with a parametric function for finding a zero of a univariate fu nction, permitting f′(x)=0 in some points. The case of multiple roots is n ot treated. The methods are proved to be quadratically convergent provided the w eak condition. Thus the methods remove the severe condition f′(x)≠0. Based on the general form of the Newton-like methods, a family of new iterative meth ods with a variable parameter are developed.

LOCAL CONVERGENCE OF INEXACT NEWTON-LIKE METHOD UNDER WEAK LIPSCHITZ CONDITIONS

The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis.The obtained results compare favorably with earlier ones such as[7,13,14,18,19].A numerical example is also provided.

CONVERGENCE OF INEXACT CONIC NEWTON METHODS

A conic Newton method is attractive because it converges to a local minimizzer rapidly from any sufficiently good initial guess. However, it may be expensive to solve the conic Newton equation at each iterate. In this paper we consider an inexact conic Newton method, which solves the couic Newton equation oldy approximately and in sonm unspecified manner. Furthermore, we show that such method is locally convergent and characterizes the order of convergence in terms of the rate of convergence of the relative residuals.

An Adaptive Spectral Conjugate Gradient Method with Restart Strategy

As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initially proposed by Jiang et al.(Computational and Applied Mathematics,2021,40:174),through the utilization of a convex combination technique.And this improvement allows for an adaptive search direction by integrating a newly constructed spectral gradient-type restart strategy.Then,we develop a new spectral CGM by employing an inexact line search to determine the step size.With the application of the weak Wolfe line search,we establish the sufficient descent property of the proposed search direction.Moreover,under general assumptions,including the employment of the strong Wolfe line search for step size calculation,we demonstrate the global convergence of our new algorithm.Finally,the given unconstrained optimization test results show that the new algorithm is effective.

关于Newton-like-iterative方法新的收敛性定理(英文)

用迭代法求解Newton-like法中的方程,T.J . Ypma提出Newton-like-iterative方法。在其早期的文章中,不精确牛顿法理论用来研究Newton-like-iterative方法的收敛性。与以往方法不同,今提出用不精确Newton-like法做相关的收敛性分析,所得定理更加简单,同时具有仿射不变性。

Accelerated Matrix Recovery via Random Projection Based on Inexact Augmented Lagrange Multiplier Method

In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.

GLOBAL CONVERGENCE OF THE GENERAL THREE TERM CONJUGATE GRADIENT METHODS WITH THE RELAXED STRONG WOLFE LINE SEARCH

The global convergence of the general three term conjugate gradient methods with the relaxed strong Wolfe line search is proved.

Inexact Newton method via Lanczos decomposed technique for solving box-constrained nonlinear systems

This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the pro- posed algorithm.

水平线性互补问题的一种非精确光滑牛顿算法

为了提高求解水平线性互补问题的效率,本文利用一种光滑函数,将水平线性互补问题转化为与之等价的光滑方程组,采用非精确牛顿法求解该方程组,得到了水平线性互补问题的一种非精确光滑牛顿算法.在适当的条件下证明了该算法的适定性和局部二阶收敛性,数值实验表明该算法稳定有效.

ADAPTIVE REGULARIZED QUASI-NEWTON METHOD USING INEXACT FIRST-ORDER INFORMATION

Classical quasi-Newton methods are widely used to solve nonlinear problems in which the first-order information is exact.In some practical problems,we can only obtain approximate values of the objective function and its gradient.It is necessary to design optimization algorithms that can utilize inexact first-order information.In this paper,we propose an adaptive regularized quasi-Newton method to solve such problems.Under some mild conditions,we prove the global convergence and establish the convergence rate of the adaptive regularized quasi-Newton method.Detailed implementations of our method,including the subspace technique to reduce the amount of computation,are presented.Encouraging numerical results demonstrate that the adaptive regularized quasi-Newton method is a promising method,which can utilize the inexact first-order information effectively.

Improved hybrid iterative optimization method for seismic full waveform inversion

In full waveform inversion （FWI）, Hessian information of the misfit function is of vital importance for accelerating the convergence of the inversion; however, it usually is not feasible to directly calculate the Hessian matrix and its inverse. Although the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） or Hessian-free inexact Newton （HFN） methods are able to use approximate Hessian information, the information they collect is limited. The two methods can be interlaced because they are able to provide Hessian information for each other; however, the performance of the hybrid iterative method is dependent on the effective switch between the two methods. We have designed a new scheme to realize the dynamic switch between the two methods based on the decrease ratio （DR） of the misfit function （objective function）, and we propose a modified hybrid iterative optimization method. In the new scheme, we compare the DR of the two methods for a given computational cost, and choose the method with a faster DR. Using these steps, the modified method always implements the most efficient method. The results of Marmousi and overthrust model testings indicate that the convergence with our modified method is significantly faster than that in the L-BFGS method with no loss of inversion quality. Moreover, our modified outperforms the enriched method by a little speedup of the convergence. It also exhibits better efficiency than the HFN method.

一种非精确非光滑信赖域算法

Aravkin等人提出了求解非光滑优化问题min_(x∈R^(d))f(x)+h(x)的非光滑信赖域算法(采用f的精确梯度),其中f是连续可微函数,h是邻近有界且下半连续的真函数。文章研究当该问题中f:=1/n ∑_(i=1)^(n)f_(i)(n很大且每个分量函数fi是连续可微)时,求解这类大规模可分离非光滑优化问题的有效算法。结合非精确算法和非光滑信赖域算法的思想,提出了用非精确梯度代替精确梯度的非精确非光滑信赖域算法。与非光滑信赖域算法(采用精确梯度)相比,该算法降低了每次迭代的计算量。在一定的假设条件下,证明了算法的迭代复杂度。

GLOBAL COVERGENCE OF THE NON-QUASI-NEWTON METHOD FOR UNCONSTRAINED OPTIMIZATION PROBLEMS

In this paper, the non-quasi-Newton＇s family with inexact line search applied to unconstrained optimization problems is studied. A new update formula for non-quasi-Newton＇s family is proposed. It is proved that the constituted algorithm with either Wolfe-type or Armijotype line search converges globally and Q-superlinearly if the function to be minimized has Lipschitz continuous gradient.

GLOBAL CONVERGENCE OF A TRUST REGION ALGORITHM USING INEXACT GRADIENT FOR EQUALITY-CONSTRAINED OPTIMIZATION

A trust-region algorithm is presented for a nonlinear optimization problem of equality-constraints. The characterization of the algorithm is using inexact gradient information. Global convergence results are demonstrated where the gradient values are obeyed a simple relative error condition.

An Inexact Restoration Package for Bilevel Programming Problems

Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the con-straints is also an optimization problem. Inexact restoration methods were introduced for solving nonlinear programming problems a few years ago. They generate a sequence of, generally, infeasible iterates with intermediate iterations that consist of inexactly restored points. In this paper we present a software environment for solving bilevel program-ming problems using an inexact restoration technique without replacing the lower level problem by its KKT optimality conditions. With this strategy we maintain the minimization structure of the lower level problem and avoid spurious solutions. The environment is a user-friendly set of Fortran 90 modules which is easily and highly configurable. It is prepared to use two well-tested minimization solvers and different formulations in one of the minimization subproblems. We validate our implementation using a set of test problems from the literature, comparing different formulations and the use of the minimization solvers.

箱约束变分不等式的一种非精确半光滑算法

为提高求解箱约束变分不等式问题的效率,文章在一个互补函数的基础上,将原问题转化为与之等价的方程组,给出一种非精确半光滑算法。在该算法的每步迭代中,相应的线性方程组都采用非精确求解方法。算法的全局收敛性被证明,数值试验表明,算法对求解该类问题稳定可靠。

ON SEMILOCAL CONVERGENCE OF INEXACT NEWTON METHODS

Inexact Newton methods are constructed by combining Newton＇s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton＇s method, we obtain a different Newton-Kantorovich theorem about Newton＇s method. When the iterative method for solving the Newton equations is specified to be the splitting method, we get two estimates about the iteration steps for the special inexact Newton methods.

求解大规模混合互补问题的Newton-GMRES方法

针对大规模混合互补问题,借助Chen-Harker-Kanzow-Smale(CHKS)光滑逼近函数,将其转化为光滑方程组。与非精确牛顿法相结合,得到了一种求解混合互补问题的Newton-GMRES算法,证明了该算法的全局收敛性。数值实验结果表明,该算法可行有效。

二次规划子问题的一种非精确光滑牛顿法

二次规划子问题的求解是解决规划问题的关键。针对二次规划子问题,利用最优性条件,借助光滑逼近函数将其转化为光滑方程组,结合非精确牛顿法得到一种求解二次规划子问题的非精确光滑牛顿法。一定条件下证明其全局收敛性。数值实验表明此算法对二次规划子问题有效。

增广Lagrange方法求解二阶锥约束变分不等式问题

应用增广Lagrange方法求解了一类二阶锥约束变分不等式问题。首先,将二阶锥约束变分不等式问题转化为等价的优化问题,从而得到其不同的等价形式;其次,应用投影算子的性质,将二阶锥约束变分不等式问题转化为方程组问题,并针对方程组问题提出了增广Lagrange方法;再次,讨论了算法的全局收敛性,同时对算法的一个特殊情况进行了深入分析,并引入一类非精确牛顿法求解算法中蕴含的子问题;最后,给出3个算例的数值实验结果,验证了算法的可行性。