A new trust region method with adaptive radius

In this paper we develop a new trust region method with adaptive radius for unconstrained optimization problems. The new method can adjust the trust region radius automatically at each iteration and possibly reduces the number of solving subproblems. We investigate the global convergence and convergence rate of this new method under some mild conditions. Theoretical analysis and numerical results show that the new adaptive trust region radius is available and reasonable and the resultant trust region method is efficient in solving practical optimization problems. The work was supported in part by NSF grant CNS-0521142, USA.     

An adaptive trust region method and its convergence

In this paper, a new trust region subproblem is proposed. The trust radius in the new subproblem adjusts itself adaptively. As a result, an adaptive trust region method is constructed based on the new trust region subproblem. The local and global convergence results of the adaptive trust region method are proved. Numerical results indicate that the new method is very efficient.     

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.     

Nonmonotone adaptive trust region method

In this paper, we propose a nonmonotone adaptive trust region method for unconstrained optimization problems. This method can produce an adaptive trust region radius automatically at each iteration and allow the functional value of iterates to increase within finite iterations and finally decrease after such finite iterations. This nonmonotone approach and adaptive trust region radius can reduce the number of solving trust region subproblems when reaching the same precision. The global convergence and convergence rate of this method are analyzed under some mild conditions. Numerical results show that the proposed method is effective in practical computation.     

Nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization

We propose a nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization. Unlike traditional trust region methods, the subproblem in our method is a simple conic model, where the Hessian of the objective function is approximated by a scalar matrix. The trust region radius is adjusted with a new self-adaptive adjustment strategy which makes use of the information of the previous iteration and current iteration. The new method needs less memory and computational efforts. The global convergence and Q-superlinear convergence of the algorithm are established under the mild conditions. Numerical results on a series of standard test problems are reported to show that the new method is effective and attractive for large scale unconstrained optimization problems.     

无约束优化的自适应信赖域方法

本文对无约束优化问题提出一个自适应信赖域方法,每次迭代都充分利用前迭代点的信息自动产生一个恰当的信赖域半径,在此区域内,二次模型与原目标函数尽可能一致,避免盲目的尝试,提高了计算效率。文中在通常条件下证明了全局收敛性及局部超线性收敛结果,给出了新算法与传统信赖域方法的数值结果,证实了新方法的有效性。     

一种解无约束优化问题的新的非单调自适应信赖域方法

本文提出了一种解无约束优化问题的新的非单调自适应信赖域方法.这种方法借助于目标函数的海赛矩阵的近似数量矩阵来确定信赖域半径.在通常的条件下,给出了新算法的全局收敛性以及局部超线性收敛的结果,数值试验验证了新的非单调方法的有效性.     

一个锥模型的自适应信赖域算法及其收敛性

本文研究了无约束最优化问题的基于锥模型的自适应信赖域算法.利用理论分析得到一个新的自适应信赖域半径.算法在每步迭代中以变化的速率、当前迭代点的信息以及水平向量信息调节信赖域半径的大小.从理论上证明了新算法的全局收敛性和Q-二阶收敛性.用数值试验验证了新算法的有效性.推广了已有的自适应信赖域算法的可行性和有效性.     

A new modified nonmonotone adaptive trust region method for unconstrained optimization

In this paper, we present an adaptive trust region method for solving unconstrained optimization problems which combines nonmonotone technique with a new update rule for the trust region radius. At each iteration, our method can adjust the trust region radius of related subproblem. We construct a new ratio to adjust the next trust region radius which is different from the ratio in the traditional trust region methods. The global and superlinear convergence results of the method are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems.     

采用不定Dogleg路径的非单调自适应信赖域算法解无约束极小化(英文)

In this paper, we combine the nonmonotone and adaptive techniques with trust region method for unconstrained minimization problems. We set a new ratio of the actual descent and predicted descent. Then, instead of the monotone sequence, the nonmonotone sequence of function values are employed. With the adaptive technique, the radius of trust region △k can be adjusted automatically to improve the efficiency of trust region methods. By means of the Bunch-Parlett factorization, we construct a method with indefinite dogleg path for solving the trust region subproblem which can handle the indefinite approximate Hessian Bk. The convergence properties of the algorithm are established. Finally, detailed numerical results are reported to show that our algorithm is efficient.     

A new nonmonotone adaptive trust region method based on simple quadratic models

Based on simple quadratic models of the trust region subproblem, we combine the trust region method with the nonmonotone and adaptive techniques to propose a new nonmonotone adaptive trust region algorithm for unconstrained optimization. Unlike traditional trust region method, our trust region subproblem is very simple by using a new scale approximation of the minimizing function??s Hessian. The new method needs less memory capacitance and computational complexity. The convergence results of the method are proved under certain conditions. Numerical results show that the new method is effective and attractive for large scale unconstrained problems.     

A new trust region method for unconstrained optimization

In this paper, we propose a new trust region method for unconstrained optimization problems. The new trust region method can automatically adjust the trust region radius of related subproblems at each iteration and has strong global convergence under some mild conditions. We also analyze the global linear convergence, local superlinear and quadratic convergence rate of the new method. Numerical results show that the new trust region method is available and efficient in practical computation.     

On memory gradient method with trust region for unconstrained optimization

In this paper we present a new memory gradient method with trust region for unconstrained optimization problems. The method combines line search method and trust region method to generate new iterative points at each iteration and therefore has both advantages of line search method and trust region method. It sufficiently uses the previous multi-step iterative information at each iteration and avoids the storage and computation of matrices associated with the Hessian of objective functions, so that it is suitable to solve large scale optimization problems. We also design an implementable version of this method and analyze its global convergence under weak conditions. This idea enables us to design some quick convergent, effective, and robust algorithms since it uses more information from previous iterative steps. Numerical experiments show that the new method is effective, stable and robust in practical computation, compared with other similar methods.     

A nonmonotone adaptive trust region method for unconstrained optimization based on conic model

In this paper, we present a nonmonotone adaptive trust region method for unconstrained optimization based on conic model. The new method combines nonmonotone technique and a new way to determine trust region radius at each iteration. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments show that our algorithm is effective.     

Convergence of supermemory gradient method

In this paper we consider the global convergence of a new supermemory gradient method for unconstrained optimization problems. New trust region radius is proposed to make the new method converge stably and averagely, and it will be suitable to solve large scale minimization problems. Some global convergence results are obtained under some mild conditions. Numerical results show that this new method is effective and stable in practical computation.     

解无约束优化问题的一个新的带线搜索的信赖域算法

在传统信赖域方法的基础上, 提出了求解无约束最优化问题的一个新的带线搜索的信赖域算法. 该算法采用大步长 Armijo 线搜索技术获得迭代步长, 克服了每次迭代求解信赖域子问题时计算量较大的缺点, 因而适用于求解大型的优化问题. 在适当的条件下, 我们证明了算法的全局收敛性. 数值实验结果表明本文所提出的算法是有效的.     

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.     

基于信赖域技术的非单调超记忆梯度算法

基于信赖域技术和修正拟牛顿方程,结合Zhang H.C.非单调策略,设计了新的求解无约束最优化问题的非单调超记忆梯度算法,分析了算法的收敛性和收敛速度.数值实验表明算法是有效的,适于求解大规模问题.     

The convergence of subspace trust region methods

The trust region method is an effective approach for solving optimization problems due to its robustness and strong convergence. However, the subproblem in the trust region method is difficult or time-consuming to solve in practical computation, especially in large-scale problems. In this paper we consider a new class of trust region methods, specifically subspace trust region methods. The subproblem in these methods has an adequate initial trust region radius and can be solved in a simple subspace. It is easier to solve than the original subproblem because the dimension of the subproblem in the subspace is reduced substantially. We investigate the global convergence and convergence rate of these methods.     

An adaptive approach of conic trust-region method for unconstrained optimization problems

In this paper, an adaptive trust region method based on the conic model for unconstrained optimization problems is proposed and analyzed. We establish the global and superlinear convergence results of the method. Numerical tests are reported that confirm the efficiency of the new method.