Gradient methods exploiting spectral properties

We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational Optimization and Applications, 2006, 33(1): 73–88) is applied for minimizing quadratic objective functions. Based on this spectral property, we develop a monotone gradient method that takes a certain number of steps using the asymptotically optimal stepsize by Dai and Yang, and then follows by some short steps associated with this new stepsize. By employing one step retard of the asymptotic optimal stepsize, a nonmonotone variant of this method is also proposed. Under mild conditions, R-linear convergence of the proposed methods is established for minimizing quadratic functions. In addition, by combining gradient projection techniques and adaptive nonmonotone line search, we further extend those methods for general bound constrained optimization. Two variants of gradient projection methods combining with the Barzilai-Borwein stepsizes are also proposed. Our numerical experiments on both quadratic and bound constrained optimization indicate that the new proposed strategies and methods are very effective.     

Trust-region method for box-constrained semismooth equations and its applications to complementary problems

In this paper, we propose a new trust-region algorithm for bound-constrained semismooth systems of equations. Trust-region subproblem is defined by minimizing a quadratic function subject only to a rectangular constraint. By employing a new active set and nonmonotone techniques, solution of the equations can be found effective. Global and local convergence results of the proposed algorithm are established under reasonable conditions. The algorithm is applied and tested on complementary problems and the experiments show that our method is efficient.     

An affine scaling interior trust-region method combining with nonmonotone line search filter technique for linear inequality constrained minimization

This paper proposes an affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints. Based on a Newton step which is derived from the complementarity conditions of linear inequality constrained optimization, a trust-region subproblem subject only to an ellipsoidal constraint is defined by minimizing a quadratic model with an appropriate quadratic function and scaling matrix. The nonmonotone schemes combining with trust-region strategy and line search filter technique can bring about speeding up the convergence progress in the case of high nonlinear. A new backtracking relevance condition is given which assures global convergence without using the switching condition used in the traditional line search filter technique. The fast local convergence rate of the proposed algorithm is achieved which is not depending on any external restoration procedure. The preliminary numerical experiments are reported to show effectiveness of the proposed algorithm.     

An adaptive nonmonotone truncated Newton method for optimal control of a class of parabolic distributed parameter systems

A black-box method using the finite elements, the Crank–Nicolson and a nonmonotone truncated Newton (TN) method is presented for solving optimal control problems (OCPs) governed by partial differential equations (PDEs). The proposed method finds the optimal control of a class of linear and nonlinear parabolic distributed parameter systems with a quadratic cost functional. To this end, the piecewise linear finite elements method and the well-known Crank–Nicolson method are used for discretizing in space and in time, respectively. Afterwards, regarding the implicit function theorem (IFT), the optimal control problem is transformed into an unconstrained nonlinear optimization problem. Considering that in a gradient-based method for solving optimal control problems, the evaluations of gradients and Hessians of the cost functional is important, hence, an adjoint technique is used to evaluate them effectively. In addition, to make a globalization strategy, we first introduce an adaptive nonmonotone strategy which properly controls the degree of nonmonotonicity and then incorporate it into an inexact Armijo-type line search approach to construct a more relaxed line search procedure. Finally, the obtained unconstrained nonlinear optimization problem is solved by utilizing the proposed nonmonotone truncated Newton method. Results gained from the new offered method compared with existing methods show that the new method is promising.     

Nonmonotone trust region methods with curvilinear path in unconstrained optimization

A general nonmonotone trust region method with curvilinear path for unconstrained optimization problem is presented. Although this method allows the sequence of the objective function values to be nonmonotone, convergence properties similar to those for the usual trust region methods with curvilinear path are proved under certain conditions. Some numerical results are reported which show the superiority of the nonmonotone trust region method with respect to the numbers of gradient evaluations and function evaluations.     

Deterministic convergence of an online gradient method for BP neural networks

Online gradient methods are widely used for training feedforward neural networks. We prove in this paper a convergence theorem for an online gradient method with variable step size for backward propagation (BP) neural networks with a hidden layer. Unlike most of the convergence results that are of probabilistic and nonmonotone nature, the convergence result that we establish here has a deterministic and monotone nature.     

A nonmonotone ODE-based method for unconstrained optimization

This paper presents a new hybrid algorithm for unconstrained optimization problems, which combines the idea of the IMPBOT algorithm with the nonmonotone line search technique. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step, via a modified limited-memory BFGS two loop recursion that requires only matrix–vector products, thus reducing the computations and storage. Furthermore, when the trial step is not accepted, the proposed method performs a line search along it using a modified nonmonotone scheme, thus a larger stepsize can be yielded in each line search procedure. Under some reasonable assumptions, the convergence properties of the proposed algorithm are analysed. Numerical results are also reported to show the efficiency of this proposed method.     

A nonmonotone adaptive trust region method and its convergence

In this paper, we combine the new trust region subproblem proposed in [1] with the nonmonotone technique to propose a new algorithm for unconstrained optimization—the nonmonotone adaptive trust region method. The local and global convergence properties of the nonmonotone adaptive trust region method are proved. Its efficiency is tested by numerical results.     

Nonmonotone Levenberg�CMarquardt training of recurrent neural architectures for processing symbolic sequences

In this paper, we present nonmonotone variants of the Levenberg–Marquardt (LM) method for training recurrent neural networks (RNNs). These methods inherit the benefits of previously developed LM with momentum algorithms and are equipped with nonmonotone criteria, allowing temporal increase in training errors, and an adaptive scheme for tuning the size of the nonmonotone slide window. The proposed algorithms are applied to training RNNs of various sizes and architectures in symbolic sequence-processing problems. Experiments show that the proposed nonmonotone learning algorithms train more effectively RNNs for sequence processing than the original monotone methods.     

Inexact quasi-Newton global convergent method for solving constrained nonsmooth equations

In this work we introduce a new method for solving nonsmooth equations with simple constraints. The method is based on the inexact and quasi-Newton approaches with backtracking strategy. Some conditions are given that ensure global superlinear convergence to a solution of the equation. We also propose a nonmonotone algorithm scheme. Both versions of the algorithm were constructed for Lipschitz continuous equations.     

线性约束优化的一类非单调信赖域算法

§1.引 言 考虑线性约束优化问题:min.f(x)s.t. aiTx=bi,i∈E,(1.1)aiTx≥bi,i∈I,其中f(x)是可行域X={x∈Rn|aiTx=bi,i∈E;aiTx≥bi,i∈I}上的连续可微函数. 多年来,问题(1.1)一直受到许多研究人员的广泛注意,相继提出了有效集方法、投影梯度法[1,2]等.特别是近几年来,信赖域方法因具有强适性、强收敛性受到更多的重视[3,8,11,12],这些方法都具有一个共同的性质:下降性,即要求在迭代点,目标函数值严格单调下降,放     

A nonmonotone scaled conjugate gradient algorithm for large-scale unconstrained optimization

This paper proposes a nonmonotone scaled conjugate gradient algorithm for solving large-scale unconstrained optimization problems, which combines the idea of scaled memoryless Broyden–Fletcher–Goldfarb–Shanno preconditioned conjugate gradient method with the nonmonotone technique. An attractive property of the proposed method is that the search direction always provides sufficient descent step at each iteration. This property is independent of the line search used. Under appropriate assumptions, the method is proven to possess global convergence for nonconvex smooth functions, and R-linear convergence for strongly convex functions. Preliminary numerical results and related comparisons show the efficiency of the proposed method in practical computation.     

A nonmonotone accelerated Levenberg–Marquardt method for the -eigenvalues of symmetric tensors

The eigenvalues of tensors become more and more important in the numerical multilinear algebra. In this paper, based on the nonmonotone technique, an accelerated Levenberg–Marquardt (LM) algorithm is presented for computing the -eigenvalues of symmetric tensors, in which an LM step and an accelerated LM step are computed at each iteration. We establish the global convergence of the proposed algorithm using properties of symmetric tensors and norms. Under the local error-bound condition, the cubic convergence of the nonmonotone accelerated LM algorithm is derived. Numerical results show that this method is efficient.     

一个松弛非单调线搜索法

对于无约束优化问题,我们用松弛技术改进了一般非单调线搜索准则,建立了相应的求解算法,并证明了算法的整体收敛性.部分数值实验结果表明,这个松弛非单调算法有效.     

Smoothing projected cyclic Barzilai–Borwein method for stochastic linear complementarity problems

In this paper, we propose a smoothing projected cyclic Barzilai–Borwein (SPCBB) method for solving the expected residual minimization formulation of stochastic linear complementarity problems (SLCPs). The SPCBB method combines the smoothing techniques and the projected Barzilai–Borwein (BB) method, where the cyclic BB scheme which resues the same BB stepsize for several consecutive iterations and a nonmonotone line search that requires an average of the successive function values decreases are employed to accelerate the convergence process. Under mild conditions, we show the convergence of the proposed method to a Clarke stationary point. Preliminary numerical results of randomly generated SLCPs show that the method is promising.     

The higher-order Levenberg–Marquardt method with Armijo type line search for nonlinear equations

To save more Jacobian calculations and achieve a faster convergence rate, Yang [A higher-order Levenberg-Marquardt method for nonlinear equations, Appl. Math. Comput. 219(22)(2013), pp. 10682–10694, doi:10.1016/j.amc.2013.04.033, 65H10] proposed a higher-order Levenberg–Marquardt (LM) method by computing the LM step and another two approximate LM steps for nonlinear equations. Under the local error bound condition, global and local convergence of this method is proved by using trust region technique. However, it is clear that the last two approximate LM steps may be not necessarily a descent direction, and standard line search technique cannot be used directly to obtain the convergence properties of this higher-order LM method. Hence, in this paper, we employ the nonmonotone second-order Armijo line search proposed by Zhou [On the convergence of the modified Levenberg-Marquardt method with a nonmonotone second order Armijo type line search, J. Comput. Appl. Math. 239 (2013), pp. 152–161] to guarantee the global convergence of this higher-order LM method. Moreover, the local convergence is also preserved under the local error bound condition. Numerical results show that the new method is efficient.     

A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints

A nonmonotone trust-region method for the solution of nonlinear systems of equations with box constraints is considered. The method differs from existing trust-region methods both in using a new nonmonotonicity strategy in order to accept the current step and a new updating technique for the trust-region-radius. The overall method is shown to be globally convergent. Moreover, when combined with suitable Newton-type search directions, the method preserves the local fast convergence. Numerical results indicate that the new approach is more effective than existing trust-region algorithms.     

连续时间 Hopfield网络模型数值实现分析

讨论使用Euler方法和梯形方法在数值求解连续时间的Hopfield网络模型时,离散时间步长的选择和迭代停止条件问题.利用凸函数的定义研究了能量函数下降的条件,根据凸函数的性质分析它的共轭函数减去二次函数之差仍为凸函数的条件.分析连续时间Hopfield网络模型的收敛性证明,提出了一个广义的连续时间Hopfield网络模型.对于常用的Euler方法和梯形方法数值求数值实现连续时间Hopfield网络,讨论了离散时间步长的选择.由于梯形方法为隐式方法,分析了它的迭代求算法的停止条件.根据连续时间Hopfield网络的特点,提出改进的迭代算法,并对其进行了分析.数值实验的结果表明,较大的离散时间步长不仅加速了数值实现,而且有利于提高优化性能.     

Deterministic nonmonotone strategies for effective training of multilayer perceptrons

We present deterministic nonmonotone learning strategies for multilayer perceptrons (MLPs), i.e., deterministic training algorithms in which error function values are allowed to increase at some epochs. To this end, we argue that the current error function value must satisfy a nonmonotone criterion with respect to the maximum error function value of the M previous epochs, and we propose a subprocedure to dynamically compute M. The nonmonotone strategy can be incorporated in any batch training algorithm and provides fast, stable, and reliable learning. Experimental results in different classes of problems show that this approach improves the convergence speed and success percentage of first-order training algorithms and alleviates the need for fine-tuning problem-depended heuristic parameters.     

A cubic regularization algorithm for unconstrained optimization using line search and nonmonotone techniques

In recent years, cubic regularization algorithms for unconstrained optimization have been defined as alternatives to trust-region and line search schemes. These regularization techniques are based on the strategy of computing an (approximate) global minimizer of a cubic overestimator of the objective function. In this work we focus on the adaptive regularization algorithm using cubics (ARC) proposed in Cartis et al. [Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results, Mathematical Programming A 127 (2011), pp. 245–295]. Our purpose is to design a modified version of ARC in order to improve the computational efficiency preserving global convergence properties. The basic idea is to suitably combine a Goldstein-type line search and a nonmonotone accepting criterion with the aim of advantageously exploiting the possible good descent properties of the trial step computed as (approximate) minimizer of the cubic model. Global convergence properties of the proposed nonmonotone ARC algorithm are proved. Numerical experiments are performed and the obtained results clearly show satisfactory performance of the new algorithm when compared to the basic ARC algorithm.