非线性最小二乘问题的信赖域方法

本文提出一个新的非线性最小二乘的信赖域方法,在该方法中每个信赖域子问题只需要一次求解,而且每次迭代的一维搜索步长因子是给定的,避开一维搜索的环节,大大地提高了算法效率.文中证明了在一定的条件下算法的全局收敛性.     

张量方法在信赖域算法中的应用

信赖域算法是求解无约束优化问题的一种有效的算法.对于该算法的子问题,本文将原来目标函数的二次模型扩展成四次张量模型,提出了一个带信赖域约束的四次张量模型优化问题的求解算法.该方法的最大特点是:不仅在张量模型的非稳定点可以得到下降方向及相应的迭代步长,而且在非局部极小值点的稳定点也可以得到下降方向及相应的迭代步长,从而在算法产生的迭代点列中存在一个子列收敛到信赖域子问题的局部极小值点.     

带有固定步长的非单调自适应信赖域算法

提出了求解无约束优化问题带有固定步长的非单调自适应信赖域算法.信赖域半径的修正采用自适应技术,算法在试探步不被接受时,采用固定步长寻找下一迭代点.并在适当的条件下,证明算法具有全局收敛性和超线性收敛性.初步的数值试验表明算法对高维问题具有较好的效果.     

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...     

一类带非单调线搜索的非单调信赖域新算法

文章结合非单调信赖域方法和非单调线搜索技术提出了一类新的无约束优化算法.与传统的非单调信赖与算法相比,此算法在每步都采用非单调Wolfe线搜索得到下一个迭代点,信赖域半径由子问题的近似解和线搜索的步长调节,这样得到的新算法不仅不需重解子问题,而且在每步迭代保证目标函数的近似海赛矩阵的正定性,在一定条件下证明了算法具有全局收敛性和Q-二次收敛性.数值试验表明算法是十分有效的.     

基于信赖域技术的处理带线性约束优化的内点算法

基于信赖域技术,本文提出了一个求解带线性等式和非负约束优化问题的内点算法,其特点是:为了求得搜索方向,算法在每一步迭代时仅需要求解一线性方程组系统,从而避免了求解带信赖域界的子问题,然后利用非精确的Armijo线搜索法来得到下一个迭代内点. 从数值计算的观点来看,这种技巧可减少计算量.在适当的条件下,文中还证明了该算法所产生的迭代序列的每一个聚点都是原问题的KKT点.     

一种求解非线性互补问题的filter内点算法

利用Armijio条件和信赖域方法,构造新的价值函数.首次将内点算法与filter技术结合起来,提出一种求解非线性互补问题的新算法,即filter内点算法.在主算法中使用Armijio型线搜索求取步长,在修复算法中使用信赖域方法进行适当控制以保证算法的收敛性.文章还讨论了算法的全局收敛性.最后用数值实验表明了该方法是有效的.     

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.     

一类带线搜索的非单调信赖域新算法

结合非单调信赖域方法,和非单调线搜索技术,提出了一种新的无约束优化算法.信赖域方法的每一步采用线搜索,使得迭代每一步都充分下降加快了迭代速度.在一定条件下,证明了算法具有全局收敛性和局部超线性.收敛速度.数值试验表明算法是十分有效的.     

一类非单调曲线搜索方法及其收敛性

本文提出一类求解无约束优化问题的非单调曲线搜索方法, 在较弱条件下证明了其收敛性.该算法有如下特点:(1)采用曲线搜索方法, 在每步迭代时同时确定下降方向和步长;(2)采用非单调搜索技巧, 产生较大的迭代步长, 降低了算法的计算量;(3)利用当前和前面迭代点的信息产生下降方向, 无需计算和存储矩阵, 适于求解大型优化问题.     

A nonmonotone trust region method with new inexact line search for unconstrained optimization

In this paper, a new nonmonotone inexact line search rule is proposed and applied to the trust region method for unconstrained optimization problems. In our line search rule, the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value, instead of the current objective function value . We can obtain a larger stepsize in each line search procedure and possess nonmonotonicity when incorporating the nonmonotone term into the trust region method. Unlike the traditional trust region method, the algorithm avoids resolving the subproblem if a trial step is not accepted. Under suitable conditions, global convergence is established. Numerical results show that the new method is effective for solving unconstrained optimization problems.     

Combining trust region and linesearch algorithm for equality constrained optimization

In this paper, a combining trust region and line search algorithm for equality constrained optimization is proposed. At each iteration, we only need to solve the trust region subproblem once, when the trust region trial step can not be accepted, we switch to line search to obtain the next iteration. Hence, the difficulty of repeated solving trust region subproblem in an iterate is avoided. In order to allow the direction of negative curvature, we add second correction step in trust region step and employ nommonotone technique in line search. The global convergence and local superlinearly rate are established under certain assumptions. Some numerical examples are given to illustrate the efficiency of the proposed algorithm.     

An effective adaptive trust region algorithm for nonsmooth minimization

In this paper, an adaptive trust region algorithm that uses Moreau–Yosida regularization is proposed for solving nonsmooth unconstrained optimization problems. The proposed algorithm combines a modified secant equation with the BFGS update formula and an adaptive trust region radius, and the new trust region radius utilizes not only the function information but also the gradient information. The global convergence and the local superlinear convergence of the proposed algorithm are proven under suitable conditions. Finally, the preliminary results from comparing the proposed algorithm with some existing algorithms using numerical experiments reveal that the proposed algorithm is quite promising for solving nonsmooth unconstrained optimization problems.     

A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization

This paper proposes a primal-dual interior point method for solving large scale nonlinearly constrained optimization problems. To solve large scale problems, we use a trust region method that uses second derivatives of functions for minimizing the barrier-penalty function instead of line search strategies. Global convergence of the proposed method is proved under suitable assumptions. By carefully controlling parameters in the algorithm, superlinear convergence of the iteration is also proved. A nonmonotone strategy is adopted to avoid the Maratos effect as in the nonmonotone SQP method by Yamashita and Yabe. The method is implemented and tested with a variety of problems given by Hock and Schittkowskis book and by CUTE. The results of our numerical experiment show that the given method is efficient for solving large scale nonlinearly constrained optimization problems.Acknowledgement The authors would like to thank anonymous referees for their valuable comments to improve the paper.     

带非线性不等式约束优化问题的信赖域算法

借助于KKT条件和NCP函数,提出了求解带非线性不等式约束优化问题的信赖域算法.该算法在每一步迭代时,不必求解带信赖域界的二次规划子问题,仅需求一线性方程组系统.在适当的假设条件下,它还是整体收敛的和局部超线性收敛的.数值实验结果表明该方法是有效的.     

A nonmonotone PSB algorithm for solving unconstrained optimization

PSB (Powell-Symmetric-Broyden) algorithm is a very important algorithm and has been extensively used in trust region methods. However, there are few studies on the line search type PSB algorithm. The primary reason is that the direction generated by this class of algorithms is not necessarily a descent direction of the objective function. In this paper, by combining a nonmonotone line search technique with the PSB method, we propose a nonmonotone PSB algorithm for solving unconstrained optimization. Under proper conditions, we establish global convergence and superlinear convergence of the proposed algorithm. At the same time we verify the efficiency of the proposed algorithm by some numerical experiments.     

INTERIOR POINT PROJECTED REDUCED HESSIAN METHOD WITH TRUST REGION STRATEGY FOR NONLINEAR CONSTRAINED OPTIMIZATION

§1　IntroductionIn this paper we analyze an interior point scaling projected reduced Hessian methodwith trust region strategy for solving the nonlinear equality constrained optimizationproblem with nonnegative constraints on variables:min　f(x)s.t.　c(x) =0 (1.1)x≥0where f∶Rn→R is the smooth nonlinear function,notnecessarily convex and c(x)∶Rn→Rm(m≤n) is the vector nonlinear function.There are quite a few articles proposing localsequential quadratic programming reduced Hessian methods…     

A hybrid trust region algorithm for unconstrained optimization

This paper presents a hybrid trust region algorithm for unconstrained optimization problems. It can be regarded as a combination of ODE-based methods, line search and trust region techniques. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step. Further, when the trial step is not accepted, the method performs an inexact line search along it instead of resolving a new linear system. Under reasonable assumptions, the algorithm is proven to be globally and superlinearly convergent. Numerical results are also reported that show the efficiency of this proposed method.     

A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC^1 CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, a new trust region algorithm for nonlinear equality constrained LC^1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subproblem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.