有界约束半光滑方程组的非单调投影信赖域方法

投影信赖域策略结合非单调线搜索算法解有界约束非线性半光滑方程组.基于简单有界约束的非线性优化问题构建信赖域子问题,半光滑类牛顿步在可行域投影得到投影牛顿的试探步,获得新的搜索方向,结合非单调线搜索技术得到回代步,获得新的步长.在合理的条件下,证明算法不仅具有整体收敛性且保持超线性收敛速率.引入非单调技术能克服高度非线性的病态问题,加速收敛性进程,得到超线性收敛速率.     

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.     

解非线性互补问题的非单调可行SQP方法

非线性互补问题可以转化成非线性约束优化问题. 提出一种非单调线搜索的可行SQP方法. 利用QP子问题的K-T点得到一个可行下降方向,通过引入一个高阶校正步以克服Maratos效应. 同时,算法采用非单调线搜索技巧获得搜索步长. 证明全局收敛性时不需要严格互补条件, 最后给出数值试验.     

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.     

一类新的非单调记忆梯度法及其全局收敛性

在非单调Armijo线搜索的基础上提出一种新的非单调线搜索,研究了一类在该线搜索下的记忆梯度法,在较弱条件下证明了其全局收敛性。与非单调Armijo线搜索相比,新的非单调线搜索在每次迭代时可以产生更大的步长,从而使目标函数值充分下降,降低算法的计算量。     

新非单调线搜索规则的Lampariello修正对角稀疏拟牛顿算法

本文设计了求解无约束最优化问题的新的非单调线搜索规则的Lampariello修正对角稀疏拟牛顿算法.新的步长规则类似于Grippo非单调线搜索规则并包含Grippo非单调线搜索规则作为特例.新的步长规则在每一次线搜索时得到一个相对于Grippo非单调线搜索规则的较大步长,同时保证算法的全局收敛性.数值例子表明算法是有效的,适合求解大规模问题.     

求解伪单调平衡问题的强收敛算法

本文研究求解实希尔伯特空间中平衡问题的算法.结合惯性方法和非单调步长,一种求解平衡问题的算法被提出,所提算法无需知道双边函数的利普希茨常数.在双边函数伪单调和满足利普希茨条件下,算法的强收敛性被证明.数值实验表明了算法的优势.     

求解伪单调变分不等式的两种投影算法

本文提出了两种求解伪单调变分不等式的定步长的投影算法．这与Solodov ＆ Tseng(1996)和He(1997)的变步长策略不同．我们证明了算法的全局收敛性，并且还在一定条件下证明了算法的Q-线性收敛性．     

一类锥模型非单调信赖域算法及收敛性分析

本文对无约束优化问题提出了一类基于锥模型的非单调信赖域算法.二次模型非单调信赖域算法是新算法的特例.在适当的条件下,证明了算法的全局收敛性及Q-二次收敛性.     

非单调多步曲线搜索方法的收敛性

提出一种求解无约束优化问题的非单调多步曲线搜索方法.此方法具有如下特点:(1)算法在产生下一个迭代点时不仅利用了当前迭代点的信息,而且还可能利用前m个迭代点的信息.这就是多步法;(2)下降方向和步长同时确定,而不是先找到方向,再由线性搜索寻找步长.这就是曲线搜索技术;(3)采用非单调搜索技巧.在较弱的条件下,我们证明了此方法的收敛性.     

An Adaptive Nonmonotone Projected Barzilai-Borwein Gradient Method with Active Set Prediction for Nonnegative Matrix Factorization

In this paper, we first present an adaptive nonmonotone term to improve the efficiency of nonmonotone line search, and then an active set identification technique is suggested to get more efficient descent direction such that it improves the local convergence behavior of algorithm and decreases the computation cost. By means of the adaptive nonmonotone line search and the active set identification technique, we put forward a global convergent gradient-based method to solve the nonnegative matrix factorization (NMF) based on the alternating nonnegative least squares framework, in which we introduce a modified Barzilai-Borwein (BB) step size. The new modified BB step size and the larger step size strategy are exploited to accelerate convergence. Finally, the results of extensive numerical experiments using both synthetic and image datasets show that our proposed method is efficient in terms of computational speed.     

A line search trust-region algorithm with nonmonotone adaptive radius for a system of nonlinear equations

In this paper, a trust-region procedure is proposed for the solution of nonlinear equations. The proposed approach takes advantages of an effective adaptive trust-region radius and a nonmonotone strategy by combining both of them appropriately. It is believed that selecting an appropriate adaptive radius based on a suitable nonmonotone strategy can improve the efficiency and robustness of the trust-region frameworks as well as decrease the computational cost of the algorithm by decreasing the required number subproblems that must be solved. The global convergence and the local Q-quadratic convergence rate of the proposed approach are proved. Preliminary numerical results of the proposed algorithm are also reported which indicate the promising behavior of the new procedure for solving the nonlinear system.     

变步长非单调模式搜索法

A varied steplength nonmonotone pattern search method is proposed in this paper. The varied steplength search strategy is designed in this method such that the pattern direction is more approximated to efficient descent direction. The interpolation and nonmonotone technique are used for improving local search and global convergence. The theoretical and numerical results show that this method is an efficient direct search method.     

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 Self-adaptive Levenberg–Marquardt Approach for Solving Systems of Nonlinear Equations

The well-known Levenberg–Marquardt method is used extensively to solve systems of nonlinear equations. An extension of the Levenberg–Marquardt method based on new nonmonotone technique is described. To decrease the total number of iterations, this method allows the sequence of objective function values to be nonmonotone, especially in the case where the objective function is ill-conditioned. Moreover, the parameter of Levenberg–Marquardt is produced according to the new nonmonotone strategy to use the advantages of the faster convergence of the Gauss–Newton method whenever iterates are near the optimizer, and the robustness of the steepest descent method in the case in which iterates are far away from the optimizer. The global and quadratic convergence of the proposed method is established. The results of numerical experiments are reported.     

A BFGS trust-region method with a new nonmonotone technique for nonlinear equations

In this paper, we consider a trust-region method for solving nonlinear equations which employs a new nonmonotone technique. A strong nonmonotone strategy and a weaker nonmonotone strategy can be obtained by choosing the parameter adaptively. Thus, the disadvantages of the traditional nonmonotone strategy can be avoided. It does not need to compute the Jacobian matrix at every iteration, so that the workload and time are decreased. Theoretical analysis indicates that the new algorithm preserves the global convergence under classical assumptions. Moreover, superlinear and quadratic convergence are established under suitable conditions. Numerical experiments show the efficiency and effectiveness of the proposed method for solving nonlinear equations.     

The convergence of conjugate gradient method with nonmonotone line search

The conjugate gradient method is a useful and powerful approach for solving large-scale minimization problems. Liu and Storey developed a conjugate gradient method, which has good numerical performance but no global convergence under traditional line searches such as Armijo line search, Wolfe line search, and Goldstein line search. In this paper we propose a new nonmonotone line search for Liu-Storey conjugate gradient method (LS in short). The new nonmonotone line search can guarantee the global convergence of LS method and has a good numerical performance. By estimating the Lipschitz constant of the derivative of objective functions in the new nonmonotone line search, we can find an adequate step size and substantially decrease the number of functional evaluations at each iteration. Numerical results show that the new approach is effective in practical computation.     

Global Convergence of Algorithms with Nonmonotone Line Search Strategy in Unconstrained Optimization

In this paper we state some nonmonotone line search strategies for unconstrained optimization algorithms. Abstracting from the concrete line search strategy we prove two general convergence results. Using this theory we can show the global convergence of the BFGS method with nonmonotone line search strategy. In contrast to some former results about nonmonotone line search strategies, both our convergence results and their proofs are natural generalizations of known results for the monotone case.     

A NEW TRUST-REGION ALGORITHM FOR FINITE MINIMAX PROBLEM

In this paper, a new trust region algorithm for minimax optimization problems is proposed, which solves only one quadratic subproblem based on a new approximation model at each iteration. The approach is different from the traditional algorithms that usually require to solve two quadratic subproblems. Moreover, to avoid Maratos effect, the nonmonotone strategy is employed. The analysis shows that, under standard conditions, the algorithm has global and superlinear convergence. Preliminary numerical experiments are conducted to show the efficiency of the new method.     

On the Nonmonotone Line Search

The technique of nonmonotone line search has received many successful applications and extensions in nonlinear optimization. This paper provides some basic analyses of the nonmonotone line search. Specifically, we analyze the nonmonotone line search methods for general nonconvex functions along different lines. The analyses are helpful in establishing the global convergence of a nonmonotone line search method under weaker conditions on the search direction. We explore also the relations between nonmonotone line search and R-linear convergence assuming that the objective function is uniformly convex. In addition, by taking the inexact Newton method as an example, we observe a numerical drawback of the original nonmonotone line search and suggest a standard Armijo line search when the nonmonotone line search condition is not satisfied by the prior trial steplength. The numerical results show the usefulness of such suggestion for the inexact Newton method.