An effective trust-region-based approach for symmetric nonlinear systems

This paper presents a new trust-region procedure for solving symmetric nonlinear systems of equations having several variables. The proposed approach takes advantage of the combination of both an effective adaptive trust-region radius and a non-monotone strategy. It is believed that the selection of an appropriate adaptive radius and the application of a suitable non-monotone strategy can improve the efficiency and robustness of the trust-region framework as well as decrease the computational costs of the algorithm by decreasing the required number of subproblems to be solved. The global convergence and the quadratic convergence of the proposed approach are proved without the non-degeneracy assumption of the exact Jacobian. The preliminary numerical results of the proposed algorithm indicating the promising behaviour of the new procedure for solving nonlinear systems are also reported.     

Global convergence of a filter-trust-region algorithm for solving nonsmooth equations

In this paper, we present a new algorithm for solving nonsmooth equations, where the function is locally Lipschitzian. The algorithm attempts to combine the efficiency of filter techniques and the robustness of trust-region method. Global convergence for this algorithm is established under reasonable assumptions.     

A new cubic convergent method for solving a system of nonlinear equations

We present a new cubic convergent method for solving a system of nonlinear equations. The new method can be viewed as a modified Chebyshev's method in which the difference of Jacobian matrixes replaces three order tensor. Therefore, the new method reduces the storage and computational cost. The new method possesses the local cubic convergence as well as Chebyshev's method. A rule is deduced to ensure the descent property of the search direction, and a nonmonotone line search technique is used to guarantee the global convergence. Numerical results indicate that the new method is competitive and efficient for some classical test problems.     

A trust-region algorithm combining line search filter technique for nonlinear constrained optimization

In this paper, we propose a trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming. At current iteration, a trial step is formed as the sum of a normal step and a tangential step which is generated by trust-region subproblem and the step size is decided by interior backtracking line search together with filter methods. Then, the next iteration is determined. This is different from general trust-region methods in which the next iteration is determined by the ratio of the actual reduction to the predicted reduction. The global convergence analysis for this algorithm is presented under some reasonable assumptions and the preliminary numerical results are reported.     

Numerical solution of KKT systems in PDE-constrained optimization problems via the affine scaling trust-region approach†

A recently proposed trust-region approach for bound-constrained nonlinear equations is applied to the Karush-Kuhn-Tucker (KKT) system arising from the discretization of a class of partial differential equation (PDE)-constrained optimization problems. Two different implementations are developed that take into account the large dimension and the special structure of the problems. The linear algebra phase is analysed considering the possibility of solving the arising linear systems by either direct methods or short-recurrence iterative linear solvers. Viability of the approach is proved through several numerical experiments on large KKT systems arising from the discretization of control problems.     

A new nonmonotone filter Barzilai–Borwein method for solving unconstrained optimization problems

In this paper, a finite filter is used in the structure of the Barzilai–Browein (BB) gradient method in order to propose a new modified BB algorithm for solving large-scale unconstrained optimization problems. Our algorithm is equipped with a relaxed nonmonotone line search technique which allows the algorithm to enjoy the nonmonotonicity properties from scratch. Under some suitable conditions, the global convergence property of the new proposed algorithm is established. Numerical results on some test problems in CUTEr library show the efficiency and effectiveness of the new algorithm in practice too.     

On affine-scaling inexact dogleg methods for bound-constrained nonlinear systems

Within the framework of affine-scaling trust-region methods for bound-constrained problems, we discuss the use of an inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Then, we focus on large-scale bound-constrained systems of nonlinear equations which often arise in practical applications when some of the unknowns are naturally subject to constraints due to physical arguments. We introduce an inexact affine-scaling method for such a class of problems that employs the inexact dogleg procedure. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix that is used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given.     

Trust-region method for box-constrained semismooth equations and its applications to complementary problems

In this paper, we propose a new trust-region algorithm for bound-constrained semismooth systems of equations. Trust-region subproblem is defined by minimizing a quadratic function subject only to a rectangular constraint. By employing a new active set and nonmonotone techniques, solution of the equations can be found effective. Global and local convergence results of the proposed algorithm are established under reasonable conditions. The algorithm is applied and tested on complementary problems and the experiments show that our method is efficient.     

基于信赖域二次规划的非线性模型预测控制优化算法

针对非线性预测控制如何在有限时域内有效的求解非凸非线性规划这一关键问题, 本文采用序列二次规划方法, 将非线性规划转化为一系列二次子规划求解. 首先根据非线性规划联立方法将系统状态和控制量同时作为优化变量, 得到以控制量步长为优化变量, 只包含不等式约束的子二次规划问题, 并用它取代原SQP子规划, 减小了子问题的规模; 随后采用基于信赖域二次规划的方法求解子规划问题, 保证每次迭代的可行性; 同时采用一种能够保持SQP问题Hessian矩阵稀疏结构的更新方法, 也在一定程度上降低了算法的复杂程度.最后的仿真结果表明了该方法的有效性.     

一族避免二阶求导的四阶迭代法

从一族解非线性方程的带参数的三阶迭代法出发,推出避免计算二阶导数的迭代族.它只需计算一阶导数值,但收敛速度却更高,至少具有四阶的收敛速度,它与别的同类型方法相比具有形式简单、计算量少等特点.最后给出数值实验,从数值实验可以看出新方法是非常有效的.     

求解非线性方程组的元胞自动机方法及其全局收敛性证明

为了求得非线性方程组所有精确解,根据元胞自动机的特点构造了求解非线性方程组的全局收敛算法。在该算法中,将非线性方程组解的理论搜索空间划分为离散搜索空间,将离散搜索空间定义为元胞空间;离散搜索空间的每个点就是一个元胞,而一个元胞对应着非线性方程组的一个试探解;元胞的状态由其空间位置及位置修正量构成。将元胞空间划分为若干个非空子集,所有元胞的状态从一个非空子集转移到另一个非空子集的状态演化过程实现了元胞空间对理论搜索空间的搜索。在元胞状态演化过程中,元胞从一个状态转移到另一个状态的状态转移概率可以计算出来;元胞演化过程中的每个状态对应于有限Markov链上的一个状态。利用可归约随机矩阵的稳定性条件证明了该算法具有全局收敛性。仿真实例表明该算法是高效的。     

一种非线性系统自适应控制及其收敛性分析*

本文对基于输入输出随机梯度的非线性系统的控制律进行了收敛性分析,给出了ＳＩＳＯ控制系统收敛的充分条件,并根据该条件给出一种非线性系统自适应控制器的设计方法。     

非线性系统的直接自适应调节律的性质分析

对直接自适应调节律的性质进行分析,在一定的条件下,证明了该调节具有最省“能量”性质,并证明在直接自适应调节律的作用下,系统的输出将收敛到希望的输出值,而且相应的控制变量序列也是收敛的。     

自适应协同进化蝙蝠算法

蝙蝠算法作为一种新型元启发式进化算法,不可避免在进化过程中存在陷入局部极值的危险.为了有效提高蝙蝠算法的进化性能,提出一种自适应协同进化的蝙蝠算法(ACEBA).为保证算法具有良好的进化结构,提出采用自适应进化种群结构,使得种群结构能够依据种群多样性在集中式结构与分布式结构之间进行切换.为协调实现主种群的勘探和子种群的开采,引入优良个体解对速度和位置进行更新,并在主种群和子种群内采用相适应的更新方式,同时将原有固定参数推广到自适应变化,并对蝙蝠行为的多普勒效应进行补偿.最后对所提出的算法进行收敛性分析和仿真验证,并与相关算法进行对比分析,充分验证了算法的正确性和有效性.     

Dynamics of a fifth-order iterative method

In this paper, we study the dynamical behaviour of a two-point iterative method with order of convergence five to solve nonlinear equations in the complex plane. In fact, we complement the dynamical study started in previous works with a more systematic analysis for polynomials with at most two different roots and different multiplicities. In addition, we characterize some polynomials of degree greater or equal to 4, such that the related methods are not generally convergent. We also analyse the degrees of the rational functions associated with two-point methods when they are applied to polynomials of degree n, showing their dependence on n ² and how this fact considerably complicates the dynamical study.