Approximation theorems for total quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to prove some approximation theorems of common fixed points for countable families of total quasi-?-asymptotically nonexpansive mappings which contain several kinds of mappings as its special cases in Banach spaces. In order to get the approximation theorems, the hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article extend and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 134 (2005) 257-266], Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces, J. Approx. Theor. 149 (2007) 103-115], Li, Su [H. Y. Li, Y. F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72(2) (2010) 847-855], Su, Xu and Zhang [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3960], Wang et al. [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. 73 (2010) 2260-2270], Chang et al. [S.S. Chang, C.K. Chan, H.W. Joseph Lee, Modified block iterative algorithm for quasi-?-asymptotically nonexpansive mappings and equilibrium problem in Banach spaces, Appl. Math. Comput. 217 (2011) 7520-7530], Ofoedu and Malonza [E.U. Ofoedu, D.M. Malonza, Hybrid approximation of solutions of nonlinear operator equations and application to equation of Hammerstein-type, Appl. Math. Comput. 217 (2011) 6019-6030] and Yao et al. [Y.H. Yao, Y.C. Liou, S.M. Kang, Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings, Appl. Math. Comput. 208 (2009) 211-218].     

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.     

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.     

Improved local convergence analysis of the Gauss–Newton method under a majorant condition

We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97–118, 2008), Chen and Li (Appl Math Comput 170:686–705, 2005), Chen and Li (Appl Math Comput 324:1381–1394, 2006), Ferreira (J Comput Appl Math 235:1515–1522, 2011), Ferreira and Gonçalves (Comput Optim Appl 48:1–21, 2011), Ferreira and Gonçalves (J Complex 27(1):111–125, 2011), Li et al. (J Complex 26:268–295, 2010), Li et al. (Comput Optim Appl 47:1057–1067, 2004), Proinov (J Complex 25:38–62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines: new directions in pure, applied and computational mathematics 185–196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123–134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.     

Two modified Dai-Yuan nonlinear conjugate gradient methods

In this paper, we propose two modified versions of the Dai-Yuan (DY) nonlinear conjugate gradient method. One is based on the MBFGS method (Li and Fukushima, J Comput Appl Math 129:15–35, 2001) and inherits all nice properties of the DY method. Moreover, this method converges globally for nonconvex functions even if the standard Armijo line search is used. The other is based on the ideas of Wei et al. (Appl Math Comput 183:1341–1350, 2006), Zhang et al. (Numer Math 104:561–572, 2006) and possesses good performance of the Hestenes-Stiefel method. Numerical results are also reported. This work was supported by the NSF foundation (10701018) of China.     

A nonmonotone trust-region line search method for large-scale unconstrained optimization

We consider an efficient trust-region framework which employs a new nonmonotone line search technique for unconstrained optimization problems. Unlike the traditional nonmonotone trust-region method, our proposed algorithm avoids resolving the subproblem whenever a trial step is rejected. Instead, it performs a nonmonotone Armijo-type line search in direction of the rejected trial step to construct a new point. Theoretical analysis indicates that the new approach preserves the global convergence to the first-order critical points under classical assumptions. Moreover, superlinear and quadratic convergence are established under suitable conditions. Numerical experiments show the efficiency and effectiveness of the proposed approach for solving unconstrained optimization problems.     

Certain improvements of Newton’s method with fourth-order convergence

In this paper we present two new schemes, one is third-order and the other is fourth-order. These are improvements of second-order methods for solving nonlinear equations and are based on the method of undetermined coefficients. We show that the fourth-order method is more efficient than the fifth-order method due to Kou et al. [J. Kou, Y. Li, X. Wang, Some modifications of Newton’s method with fifth-order covergence, J. Comput. Appl. Math., 209 (2007) 146–152]. Numerical examples are given to support that the methods thus obtained can compete with other iterative methods.     

Accelerated conjugate gradient algorithm with finite difference Hessian/vector product approximation for unconstrained optimization

In this paper we propose a fundamentally different conjugate gradient method, in which the well-known parameter β_k is computed by an approximation of the Hessian/vector product through finite differences. For search direction computation, the method uses a forward difference approximation to the Hessian/vector product in combination with a careful choice of the finite difference interval. For the step length computation we suggest an acceleration scheme able to improve the efficiency of the algorithm. Under common assumptions, the method is proved to be globally convergent. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with conjugate gradient algorithms including CONMIN by Shanno and Phua [D.F. Shanno, K.H. Phua, Algorithm 500, minimization of unconstrained multivariate functions, ACM Trans. Math. Softw. 2 (1976) 87–94], SCALCG by Andrei [N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl. 38 (2007) 401–416; N. Andrei, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Methods Softw. 22 (2007) 561–571; N. Andrei, A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett. 20 (2007) 645–650], and new conjugacy condition and related new conjugate gradient by Li, Tang and Wei [G. Li, C. Tang, Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math. 202 (2007) 523–539] or truncated Newton TN by Nash [S.G. Nash, Preconditioning of truncated-Newton methods, SIAM J. on Scientific and Statistical Computing 6 (1985) 599–616] using a set of 750 unconstrained optimization test problems show that the suggested algorithm outperforms these conjugate gradient algorithms as well as TN.     

A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α$\alpha$-inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].     

A solution method to the problem proposed by Wang in voting systems

This paper proposes a method to rank multiple efficient candidates, which often happens in DEA method, by comparing the least relative total scores for each efficient candidate with the best and the least relative total scores measured in the same range. By a numerical example, our model is used to identify efficient candidate and the model can get less efficient candidates too than that can be identified by the model given by Wang and Chin [Y.M. Wang, K.S. Chin, Discriminating DEA efficient candidates by considering their least relative total scores, J. Comput. Appl. Math. 206 (2007) 209–215]. This paper also points out that there is a drawback in the theorem about ε$\epsilon$ given by Wang and Chin [Y.M. Wang, K.S. Chin, Discriminating DEA efficient candidates by considering their least relative total scores, J. Comput. Appl. Math. 206 (2007) 209–215].     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

Several existence theorems of nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales

In this paper, several existence theorems of positive solutions are established for nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, as an application, an example to demonstrate our results is given. The conditions we used in the paper are different from those in [H.R. Sun, W.T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524; H.R. Sun, W.T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica 49 (2006) 369–380 (in Chinese); Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math. Anal. Appl. 315 (2006) 144–153; Y. Wang, W. Ge, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian, Nonlinear Appl. 66 (6) (2007) 1246–1256].     

Existence theorem and algorithm for a general implicit variational inequality in Banach space

By using the generalized f-projection operator, the existence theorem of solutions for the general implicit variational inequality GIVI(T-ξ,K) is proved without assuming the monotonicity of operators in reflexive and smooth Banach space. An iterative algorithm for approximating solution of the general implicit variational inequality is suggested also, and the convergence for this iterative scheme is shown. These theorems extend the corresponding results of Wu and Huang [K.Q. Wu, N.J. Huang, Comput. Math. Appl. 54 (2007) 399–406], Wu and Huang [K.Q. Wu, N.J. Huang, Bull. Austral. Math. Soc. 73 (2006) 307–317], Zeng and Yao [L.C. Zeng, J.C. Yao, J. Optimiz. Theory Appl. 132 (2) (2007) 321–337] and Li [J. Li, J. Math. Anal. Appl. 306 (2005) 55–71].     

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 modified halpern-type iteration algorithm for totally quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to modify the Halpern-type iteration algorithm for total quasi-?-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of [X.L. Qin, Y.J. Cho, S.M. Kang, H. Y. Zhou, Convergence of a modified Halpern-type iterative algorithm for quasi-?-nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051-1055], [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3906], [C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411] and others.     

The strong convergence of a three-step algorithm for the split feasibility problem

Based on the very recent work by Dang and Gao (Invers Probl 27:1–9, 2011) and Wang and Xu (J Inequal Appl, doi:10.1155/2010/102085, 2010), and inspired by Yao (Appl Math Comput 186:1551–1558, 2007), Noor (J Math Anal Appl 251:217–229, 2000), and Xu (Invers Probl 22:2021–2034, 2006), we suggest a three-step KM-CQ-like method for solving the split common fixed-point problems in Hilbert spaces. Our results improve and develop previously discussed feasibility problem and related algorithms.     

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.     

解无约束最优化的基于锥模型的过滤集- 信赖域方法

锥模型优化方法是一类非二次模型优化方法, 它在每次迭代中比标准的二次模型方法含有更丰富的插值信息. Di 和Sun (1996) 提出了解无约束优化问题的锥模型信赖域方法. 本文根据Fletcher 和Leyffer (2002) 的过滤集技术的思想, 在Di 和Sun (1996) 工作的基础上, 提出了解无约束优化问题的基于锥模型的过滤集信赖域算法. 在适当的条件下, 我们证明了新算法的收敛性. 有限的数值试验结果表明新算法是有效的.     

解大型无约束优化的新的非单调简单模型BB-TR方法（英文）

The trust region(TR) method for optimization is a class of effective methods.The conic model can be regarded as a generalized quadratic model and it possesses the good convergence properties of the quadratic model near the minimizer.The Barzilai and Borwein(BB) gradient method is also an effective method,it can be used for solving large scale optimization problems to avoid the expensive computation and storage of matrices.In addition,the BB stepsize is easy to determine without large computational efforts.In this paper,based on the conic trust region framework,we employ the generalized BB stepsize,and propose a new nonmonotone adaptive trust region method based on simple conic model for large scale unconstrained optimization.Unlike traditional conic model,the Hessian approximation is an scalar matrix based on the generalized BB stepsize,which resulting a simple conic model.By adding the nonmonotone technique and adaptive technique to the simple conic model,the new method needs less storage location and converges faster.The global convergence of the algorithm is established under certain conditions.Numerical results indicate that the new method is effective and attractive for large scale unconstrained optimization problems.     

解新锥模型信赖域子问题的折线法

本文以新锥模型信赖域子问题的最优性条件为理论基础,认真讨论了新子问题的锥函数性质,分析了此函数在梯度方向及与牛顿方向连线上的单调性.在此基础上本文提出了一个求解新锥模型信赖域子问题折线法,并证明了这一子算法保证解无约束优化问题信赖域法全局收敛性要满足的下降条件.本文获得的数值实验表明该算法是有效的.