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Two globally convergent nonmonotone trust-region methods for unconstrained optimization  [PDF]
Masoud Ahookhosh,Susan Ghaderi
Mathematics , 2015,
Abstract: This paper addresses some trust-region methods equipped with nonmonotone strategies for solving nonlinear unconstrained optimization problems. More specifically, the importance of using nonmonotone techniques in nonlinear optimization is motivated, then two new nonmonotone terms are proposed, and their combinations into the traditional trust-region framework are studied. The global convergence to first- and second-order stationary points and local superlinear and quadratic convergence rates for both algorithms are established. Numerical experiments on the \textsf{CUTEst} test collection of unconstrained problems and some highly nonlinear test functions are reported, where a comparison among state-of-the-art nonmonotone trust-region methods show the efficiency of the proposed nonmonotne schemes.
A Nonmonotone Trust Region Algorithm Based on the Average of the Successive Penalty Function Values for Nonlinear Optimization  [PDF]
Zhensheng Yu,Jinhong Yu
ISRN Operations Research , 2013, DOI: 10.1155/2013/495378
Abstract: We present a nonmonotone trust region algorithm for nonlinear equality constrained optimization problems. In our algorithm, we use the average of the successive penalty function values to rectify the ratio of predicted reduction and the actual reduction. Compared with the existing nonmonotone trust region methods, our method is independent of the nonmonotone parameter. We establish the global convergence of the proposed algorithm and give the numerical tests to show the efficiency of the algorithm. 1. Introduction In this paper, we consider the equality constrained optimization problem as follows: where , , , , and are assumed to be twice continuously differentiable. Trust region method is one of the well-known methods for solving problem (1). Due to its strong convergence and robustness, trust region methods have been proved to be efficient for both unconstrained and constrained optimization problems [1–9]. Most traditional trust region methods are of descent type methods; namely, they accept only a trial point as the next iterate if its associated merit function value is strictly less than that of the current iterate. However, just as pointed out by Toint [10], the nonmonotone techniques are helpful to overcome the case that the sequence of iterates follows the bottom of curved narrow valleys, a common occurrence in difficult nonlinear problems. Hence many nonmonotone algorithms are proposed to solve the unconstrained and constrained optimization problems [11–20]. Numerical tests show that the performance of the nonmonotone technique is superior to those of the monotone cases. The nonmonotone technique was originally proposed by Grippo, Lampariello and Lucidi [13] for unconstrained optimization problems based on Newton’s method, in which the stepsize satisfies the following condition: where , , and is a prefixed nonnegative integer. Although the nonmonotone technique based on (2) works well in many cases, there are some drawbacks. Firstly, a good function value generated in any iteration is essentially discarded due to the maximum in (2). Secondly, in some cases, the numerical performance is heavily dependent on the choice of (see, e.g., [16, 21]). To overcome these drawback, Zhang and Hager [21] proposed another nonmonotone algorithm, and they used the average of function values to replace the maximum function value in (2). The numerical tests show that their nonmonotone line search algorithm used fewer function and gradient evaluations, on average, than either the monotone or the traditional nonmonotone scheme. Recently, Mo and Zhang [16] extended
A Trust-Region-Based BFGS Method with Line Search Technique for Symmetric Nonlinear Equations  [PDF]
Gonglin Yuan,Shide Meng,Zengxin Wei
Advances in Operations Research , 2009, DOI: 10.1155/2009/909753
Abstract: A trust-region-based BFGS method is proposed for solving symmetric nonlinear equations. In this given algorithm, if the trial step is unsuccessful, the linesearch technique will be used instead of repeatedly solving the subproblem of the normal trust-region method. We establish the global and superlinear convergence of the method under suitable conditions. Numerical results show that the given method is competitive to the normal trust region method.
A Nonmonotone Line Search Method for Symmetric Nonlinear Equations  [PDF]
Gonglin Yuan, Laisheng Yu
Intelligent Control and Automation (ICA) , 2010, DOI: 10.4236/ica.2010.11004
Abstract: In this paper, we propose a new method which based on the nonmonotone line search technique for solving symmetric nonlinear equations. The method can ensure that the search direction is descent for the norm function. Under suitable conditions, the global convergence of the method is proved. Numerical results show that the presented method is practicable for the test problems.
On an adaptive regularization for ill-posed nonlinear systems and its trust-region implementation  [PDF]
Stefania Bellavia,Benedetta Morini,Elisa Riccietti
Mathematics , 2015,
Abstract: In this paper we address the stable numerical solution of nonlinear ill-posed systems by a trust-region method. We show that an appropriate choice of the trust-region radius gives rise to a procedure that has the potential to approach a solution of the unperturbed system. This regularizing property is shown theoretically and validated numerically.
A MODIFIED TRUST-REGION METHOD WITH GLOBAL CONVERGENCE FOR SYMMETRIC NONLINEAR EQUATIONS
具有全局收敛性的求解对称非线性方程组的一个修改的信赖域方法

Yuan Gonglin,Lu Xiwen,Wei Zengxin,
袁功林
,鲁习文,韦增欣

计算数学 , 2007,
Abstract: In this paper,a modified trust-region method for solving symmetric nonlinear equations is proposed.We establish the global convergence of the presented method under favorable conditions.Some preliminary numerical results show that this method is effective for the given problems.
一种非单调自适应信赖域法
A Nonmonotonic Self-Adaptive Trust Region Algorithm
 [PDF]

杭丹, 王晓燕, 郝建忠, 王娅
Pure Mathematics (PM) , 2013, DOI: 10.12677/PM.2013.35048
Abstract:
求解无约束优化问题,本文给出了一种改进的非单调信赖域算法。采用了非单调技术,当试探步不被接受时,下一步的迭代步长由一个固定公式给出。并直接给出调整信赖域半径公式数值试验结果表明新算法的有效性。在适当的条件下,证明了算法的全局收敛性。
We propose an improved nonmonotonic trust region algorithm. Our method is to use the nonmonotone technique and if the trial step is rejected, the stepsize is computed by a fixed formula. The trust region radius is updated at a variable rate. Numerical experiment results show that the new algorithm is effective. Under mild conditions, we prove that the algorithm is global convergence.
A nonmonotone trust region algorithm for unconstrained nonsmooth optimization

科学通报(英文版) , 1996,
Abstract:
An Improved Nonmonotone Filter Trust Region Method for Equality Constrained Optimization  [cached]
Zhong Jin
Abstract and Applied Analysis , 2013, DOI: 10.1155/2013/163487
Abstract:
Trust Region Policy Optimization  [PDF]
John Schulman,Sergey Levine,Philipp Moritz,Michael I. Jordan,Pieter Abbeel
Computer Science , 2015,
Abstract: In this article, we describe a method for optimizing control policies, with guaranteed monotonic improvement. By making several approximations to the theoretically-justified scheme, we develop a practical algorithm, called Trust Region Policy Optimization (TRPO). This algorithm is effective for optimizing large nonlinear policies such as neural networks. Our experiments demonstrate its robust performance on a wide variety of tasks: learning simulated robotic swimming, hopping, and walking gaits; and playing Atari games using images of the screen as input. Despite its approximations that deviate from the theory, TRPO tends to give monotonic improvement, with little tuning of hyperparameters.
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