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.     

An efficient nonmonotone trust-region method for unconstrained optimization

The monotone trust-region methods are well-known techniques for solving unconstrained optimization problems. While it is known that the nonmonotone strategies not only can improve the likelihood of finding the global optimum but also can improve the numerical performance of approaches, the traditional nonmonotone strategy contains some disadvantages. In order to overcome to these drawbacks, we introduce a variant nonmonotone strategy and incorporate it into trust-region framework to construct more reliable approach. The new nonmonotone strategy is a convex combination of the maximum of function value of some prior successful iterates and the current function value. It is proved that the proposed algorithm possesses global convergence to first-order and second-order stationary points under some classical assumptions. Preliminary numerical experiments indicate that the new approach is considerably promising for solving unconstrained optimization problems.     

An inexact line search approach using modified nonmonotone strategy for unconstrained optimization

This paper concerns with a new nonmonotone strategy and its application to the line search approach for unconstrained optimization. It has been believed that nonmonotone techniques can improve the possibility of finding the global optimum and increase the convergence rate of the algorithms. We first introduce a new nonmonotone strategy which includes a convex combination of the maximum function value of some preceding successful iterates and the current function value. We then incorporate the proposed nonmonotone strategy into an inexact Armijo-type line search approach to construct a more relaxed line search procedure. The global convergence to first-order stationary points is subsequently proved and the R-linear convergence rate are established under suitable assumptions. Preliminary numerical results finally show the efficiency and the robustness of the proposed approach for solving unconstrained nonlinear optimization problems.     

Combining nonmonotone conic trust region and line search techniques for unconstrained optimization

In this paper, we propose a trust region method for unconstrained optimization that can be regarded as a combination of conic model, nonmonotone and line search techniques. Unlike in traditional trust region methods, the subproblem of our algorithm is the conic minimization subproblem; moreover, our algorithm performs a nonmonotone line search to find the next iteration point when a trial step is not accepted, instead of resolving the subproblem. The global and superlinear convergence results for the algorithm are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems.     

A new nonmonotone line search technique for unconstrained optimization

In this paper, we propose a new nonmonotone line search technique for unconstrained optimization problems. By using this new technique, we establish the global convergence under conditions weaker than those of the existed nonmonotone line search techniques.     

基于非单调技术的ODE型混合方法

基于非单调线搜索技术和IMPBOT算法,提出了一个求解无约束优化问题的ODE型混合方法.该方法的主要特点是：为了求得试验步,该方法在每次迭代时不必求解带信赖域界的子问题,仅需要求解一线性方程组系统;当试验步不被接受时,该方法就执行改进的Wolfe-型非单调线搜索来获得下一个新的迭代点,从而避免了反复求解线性方程组系统. 在一定条件下,所提算法还是整体收敛和超线性收敛的. 数值试验结果表明该方法是有效的.     

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.     

An affine scaling interior trust-region method combining with line search filter technique for optimization subject to bounds on variables

This paper proposes and analyzes an affine scaling trust-region method with line search filter technique for solving nonlinear optimization problems subject to bounds on variables. At the current iteration, the trial step is generated by the general trust-region subproblem which is defined by minimizing a quadratic function subject only to an affine scaling ellipsoidal constraint. Both trust-region strategy and line search filter technique will switch to trail backtracking step which is strictly feasible. Meanwhile, the proposed method does not depend on any external restoration procedure used in line search filter technique. A new backtracking relevance condition is given which is weaker than the switching condition to obtain the global convergence of the algorithm. The global convergence and fast local convergence rate of this algorithm are established under reasonable assumptions. Preliminary numerical results are reported indicating the practical viability and show the effectiveness of the proposed algorithm.     

The Global Convergence of Self-Scaling BFGS Algorithm with Nonmonotone Line Search for Unconstrained Nonconvex Optimization Problems

The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex （or even uniformly convex）. We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems.     

A nonmonotone trust-region method of conic model for unconstrained optimization

In this paper, we present a nonmonotone trust-region method of conic model for unconstrained optimization. The new method combines a new trust-region subproblem of conic model proposed in [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231] with a nonmonotone technique for solving unconstrained optimization. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the method of [Y. Ji, S.J. Qu, Y.J. Wang, H.M. Li, A conic trust-region method for optimization with nonlinear equality and inequality 4 constrains via active-set strategy, Appl. Math. Comput. 183 (2006) 217–231].     

A hybrid of adjustable trust-region and nonmonotone algorithms for unconstrained optimization

This study devotes to incorporating a nonmonotone strategy with an automatically adjusted trust-region radius to propose a more efficient hybrid of trust-region approaches for unconstrained optimization. The primary objective of the paper is to introduce a more relaxed trust-region approach based on a novel extension in trust-region ratio and radius. The next aim is to employ stronger nonmonotone strategies, i.e. bigger trust-region ratios, far from the optimizer and weaker nonmonotone strategies, i.e. smaller trust-region ratios, close to the optimizer. The global convergence to first-order stationary points as well as the local superlinear and quadratic convergence rates are also proved under some reasonable conditions. Some preliminary numerical results and comparisons are also reported.     

一种非单调滤子信赖域算法解线性不等式约束优化

本文给出了一种新的多维滤子算法结合非单调信赖域策略解线性约束优化.目标函数及其投影梯度的分量组成了新的多维滤子，并且与信赖域半径有关.当信赖域半径充分小时，新的滤子能接受试探点，避免算法无限循环.非单调信赖域策略保证了新算法的整体收敛性.目前为止，多维滤子算法局部收敛性分析仍然没有解决，在合理假设下，我们分析了新算法的局部超线性收敛性.数值结果验证了算法的有效性.     

A new nonmonotone trust-region method of conic model for solving unconstrained optimization

In this paper, we present a new nonmonotone trust-region method of conic model for solving unconstrained optimization problems. Both the local and global convergence properties are analyzed under reasonable assumptions. Numerical experiments are conducted to compare this method with some existed ones which indicate that the new method is efficient.     

A nonmonotone trust region method based on nonincreasing technique of weighted average of the successive function values

In this paper we propose a nonmonotone trust region method. Unlike traditional nonmonotone trust region method, the nonmonotone technique applied to our method is based on the nonmonotone line search technique proposed by Zhang and Hager [A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14(4) (2004) 1043–1056] instead of that presented by Grippo et al. [A nonmonotone line search technique for Newton's method, SIAM J. Numer. Anal. 23(4) (1986) 707–716]. So the method requires nonincreasing of a special weighted average of the successive function values. Global and superlinear convergence of the method are proved under suitable conditions. Preliminary numerical results show that the method is efficient for unconstrained optimization problems.     

Modified nonmonotone Armijo line search for descent method

Nonmonotone line search approach is a new technique for solving optimization problems. It relaxes the line search range and finds a larger step-size at each iteration, so as to possibly avoid local minimizer and run away from narrow curved valley. It is helpful to find the global minimizer of optimization problems. In this paper we develop a new modification of matrix-free nonmonotone Armijo line search and analyze the global convergence and convergence rate of the resulting method. We also address several approaches to estimate the Lipschitz constant of the gradient of objective functions that would be used in line search algorithms. Numerical results show that this new modification of Armijo line search is efficient for solving large scale unconstrained optimization problems.     

非单调滤子曲率线搜索算法解无约束非凸优化

宇和濮在文[Yu Z S,Pu D G.A new nonmonotone line search technique for unconstrained optimization[J].J Comput Appl Math,2008,219:134-144]中提出了一种非单调的线搜索算法解无约束优化问题.和他们的工作不同,当优化问题非凸时,本文给出了一种非单调滤子曲率线搜索算法.通过使用海森矩阵的负曲率信息,算法产生的迭代序列被证明收敛于一个满足二阶充分性条件的点.在不需要假设极限点存在的情况下,证明了算法具有整体收敛性,而且分析了该算法的收敛速率.数值试验表明算法的有效性.     

Monotone and nonmonotone trust-region-based algorithms for large scale unconstrained optimization problems

Two trust regions algorithms for unconstrained nonlinear optimization problems are presented: a monotone and a nonmonotone one. Both of them solve the trust region subproblem by the spectral projected gradient (SPG) method proposed by Birgin, Martínez and Raydan (in SIAM J. Optim. 10(4):1196?C1211, 2000). SPG is a nonmonotone projected gradient algorithm for solving large-scale convex-constrained optimization problems. It combines the classical projected gradient method with the spectral gradient choice of steplength and a nonmonotone line search strategy. The simplicity (only requires matrix-vector products, and one projection per iteration) and rapid convergence of this scheme fits nicely with globalization techniques based on the trust region philosophy, for large-scale problems. In the nonmonotone algorithm the trial step is evaluated by acceptance via a rule which can be considered a generalization of the well known fraction of Cauchy decrease condition and a generalization of the nonmonotone line search proposed by Grippo, Lampariello and Lucidi (in SIAM J. Numer. Anal. 23:707?C716, 1986). Convergence properties and extensive numerical results are presented. Our results establish the robustness and efficiency of the new algorithms.     

An adaptive nonmonotone global Barzilai–Borwein gradient method for unconstrained optimization

The Barzilai–Borwein (BB) gradient method has received many studies due to its simplicity and numerical efficiency. By incorporating a nonmonotone line search, Raydan (SIAM J Optim. 1997;7:26–33) has successfully extended the BB gradient method for solving general unconstrained optimization problems so that it is competitive with conjugate gradient methods. However, the numerical results reported by Raydan are poor for very ill-conditioned problems because the effect of the degree of nonmonotonicity may be noticeable. In this paper, we focus more on the nonmonotone line search technique used in the global Barzilai–Borwein (GBB) gradient method. We improve the performance of the GBB gradient method by proposing an adaptive nonmonotone line search based on the morphology of the objective function. We also prove the global convergence and the R-linear convergence rate of the proposed method under reasonable assumptions. Finally, we give some numerical experiments made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed method in the sense of the performance profile introduced (Math Program. 2002;91:201–213) by Dolan and Moré.     

A Nonmonotone Line Search Slackness Technique for Unconstrained Optimization

This paper mainly aims to study a new nonmonotone line search slackness technique for unconstrained optimization problems and show that it possesses the global convergence without needing condition of convexity. We establish the corresponding algorithm and illustrate its effectiveness by virtue of some numerical tests. Simulation results indicate that the proposed method is very effective for non-convex functions.     

A new family of conjugate gradient methods

In this paper we develop a new class of conjugate gradient methods for unconstrained optimization problems. A new nonmonotone line search technique is proposed to guarantee the global convergence of these conjugate gradient methods under some mild conditions. In particular, Polak–Ribiére–Polyak and Liu–Storey conjugate gradient methods are special cases of the new class of conjugate gradient methods. By estimating the local Lipschitz constant of the derivative of objective functions, we can find an adequate step size and substantially decrease the function evaluations at each iteration. Numerical results show that these new conjugate gradient methods are effective in minimizing large-scale non-convex non-quadratic functions.