A NONMONOTONE TRUST REGION METHOD FOR NONLINEAR LEAST SQUARES PROBLEMS

In this paper we present a nonmonotone trust region method for nonlinear least squares problems with zero-residual and prove its convergence properties. The extensive numerical results are reported which show that the nonmonotone trust region method is generally superior to the usual trust region method.     

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.     

A NONMONOTONE TRUST REGION ALGORITHM FOR NONLINEAR OPTIMIZATION SUBJECT TO GENERAL CONSTRAINTS

In this paper we present a nonmonotone trust region algorithm for general nonlinear constrained optimization problems. The main idea of this paper is to combine Yuan's technique[1] with a nonmonotone method similar to Ke and Han [2]. This new algorithm may not only keep the robust properties of the algorithm given by Yuan, but also have some advantages led by the nonmonotone technique. Under very mild conditions, global convergence for the algorithm is given. Numerical experiments demonstrate the efficiency of the algorithm.     

A TRUST REGION METHOD WITH A CONIC MODEL FOR NONLINEARLY CONSTRAINED OPTIMIZATION

Trust region methods are powerful and effective optimization methods.The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods.The advantages of the above two methods can be combined to form a more powerful method for constrained optimization.The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound.At the same time,the new algorithm still possesses robust global properties.The global convergence of the new algorithm under standard conditions is established.     

A New Nonmonotone Trust Region Barzilai-Borwein Method for Unconstrained Optimization Problems

In this paper,we propose a new nonmonotone trust region Barzilai-Borwein(BB for short)method for solving unconstrained optimization problems.The proposed method is given by a novel combination of a modified Metropolis criterion,BB-stepsize and trust region method.The new method uses the reciprocal of BB-stepsize to approximate the Hessian matrix of the objective function in the trust region subproblems,and accepts some bad solutions according to the modified Metropolis criterion based on simulated annealing idea.Under some suitable assumptions,the global convergence of the new method is established.Some preliminary numerical results indicate that,the new method is more efficient compared with the existing trust region BB method.     

On convergence analysis of a derivative-free trust region algorithm for constrained optimization with separable structure

In this paper,we propose a derivative-free trust region algorithm for constrained minimization problems with separable structure,where derivatives of the objective function are not available and cannot be directly approximated.At each iteration,we construct a quadratic interpolation model of the objective function around the current iterate.The new iterates are generated by minimizing the augmented Lagrangian function of this model over the trust region.The filter technique is used to ensure the feasibility and optimality of the iterative sequence.Global convergence of the proposed algorithm is proved under some suitable assumptions.     

A Nonmonotone Trust Region Algorithm for Nonlinear Optimization Subject to General Constraints

In this paperwe present a nonmonotone trust region algorthm for general nonlinear constrained optimization problems.The main idea of this paper is to combine Yuan‘‘‘‘s technique[1]with a nonmonotone method similar to Ke and Han[2].This new algorithm may not only keep the robust properties of the algorithm given by Yuan,but also have some advantages led by the nonmonotone technique.Under very mild conditions,global convergence for the algorithm is given.Numerical experiments demostrate the efficency of the algorithm.     

信赖域策略的内点投影既约Hessian方法解非线性约束优化

A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on variables is proposed. In order to deal with large problems,a pair of trust region subproblems in horizontal and vertical subspaces is used to replace the general full trust region subproblem. The horizontal trust region subproblem in the algorithm is only a general trust region subproblem while the vertical trust region subproblem is defined by a parameter size of the vertical direction subject only to an ellipsoidal constraint. Both trust region strategy and line search technique at each iteration switch to obtaining a backtracking step generated by the two trust region subproblems. By adopting the l1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion and the second order correction step are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases.     

CONIC TRUST REGION METHOD FOR LINEARLY CONSTRAINED OPTIMIZATION

In this paper we present a trust region method of conic model for linearly constrained optimization problems.We discuss trust region approaches with conic model subproblems.Some equivalent variation properties and optimality conditions are given.A trust region algorithm based on conic model is constructed.Global convergence of the method is established.     

An algorithm for solving new trust region subproblem with conic model

The new trust region subproblem with the conic model was proposed in 2005, and was divided into three different cases. The first two cases can be converted into a quadratic model or a convex problem with quadratic constraints, while the third one is a nonconvex problem. In this paper, first we analyze the nonconvex problem, and reduce it to two convex problems. Then we discuss some dual properties of these problems and give an algorithm for solving them. At last, we present an algorithm for solving the new trust region subproblem with the conic model and report some numerical examples to illustrate the efficiency of the algorithm.     

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.     

一类新的非单调信赖域算法

提出了一类带线性搜索的非单调信赖域算法.算法将非单调Armijo线性搜索技术与信赖域方法相结合,使算法不需重解子问题.而且由于采用了MBFGS校正公式,使矩阵Bk能较好地逼近目标函数的Hesse矩阵并保持正定传递.在较弱的条件下,证明了算法的全局收敛性.数值结果表明算法是有效的.     

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.     

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.     

一个解非线性方程组新的非单调自适应信赖域法

本文利用函数平均权重的非单调技术以及自适应信赖域方法,提出一个解非线性方程组的非单调自适应信赖域法.并在适当假设条件下,讨论了算法的全局收敛性.数值试验表明了算法是有效的.     

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.     

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...     

无约束最优化的信赖域BB法

梯度法是求解无约束最优化的一类重要方法.步长选取的好坏与梯度法的数值表现息息相关.注意到BB步长隐含了目标函数的二阶信息,本文将BB法与信赖域方法相结合,利用BB步长的倒数去近似目标函数的Hesse矩阵,同时利用信赖域子问题更加灵活地选取梯度法的步长,给出求解无约束最优化问题的单调和非单调信赖域BB法.在适当的假设条件下,证明了算法的全局收敛性.数值试验表明,与已有的求解无约束优化问题的BB类型的方法相比,非单调信赖域BB法中e_k=‖x_k-x~*‖的下降呈现更明显的阶梯状和单调性,因此收敛速度更快.     

A nonmonotone conic trust region method based on line search for solving unconstrained optimization

In this paper, we present a nonmonotone conic trust region method based on line search technique for unconstrained optimization. The new algorithm can be regarded as a combination of nonmonotone technique, line search technique and conic trust region method. When a trial step is not accepted, the method does not resolve the trust region subproblem but generates an iterative point whose steplength satisfies some line search condition. The function value can only be allowed to increase when trial steps are not accepted in close succession of iterations. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the existing methods.