Nonmonotone filter DQMM method for the system of nonlinear equations

In this paper, we propose a nonmonotone filter Diagonalized Quasi-Newton Multiplier (DQMM) method for solving system of nonlinear equations. The system of nonlinear equations is transformed into a constrained nonlinear programming problem which is then solved by nonmonotone filter DQMM method. A nonmonotone criterion is used to speed up the convergence progress in some ill-conditioned cases. Under reasonable conditions, we give the global convergence properties. The numerical experiments are reported to show the effectiveness of the proposed algorithm.     

求解带界约束的非线性方程组的混合方法

基于非单调技术和L-M算法, 提出了一种新的求解带界约束的非线性方程组的混合方法. 在一定条件下, 该算法具有全局收敛性. 数值试验表明该算法是有效的.     

求解非线性互补问题的一类光滑Broyden-like方法

非线性互补问题可以等价地转换为光滑方程组来求解. 基于一种新的非单调线搜索准则, 提出了求解非线性互补问题等价光滑方程组的一类新的非单调光滑 Broyden-like 算法.在适当的假设条件下, 证明了该算法的全局收敛性与局部超线性收敛性. 数值实验表明所提出的算法是有效的.     

A modified SQP algorithm for minimax problems

In this paper, a modified nonmonotone line search SQP algorithm for nonlinear minimax problems is presented. During each iteration of the proposed algorithm, a main search direction is obtained by solving a reduced quadratic program (QP). In order to avoid the Maratos effect, a correction direction is generated by solving the reduced system of linear equations. Under mild conditions, the global and superlinear convergence can be achieved. Finally, some preliminary numerical results are reported.     

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.     

Nonmonotone smoothing Broyden-like method for generalized nonlinear complementarity problems

Based on a new symmetrically perturbed smoothing function, the generalized nonlinear complementarity problem defined on a polyhedral cone is reformulated as a system of smoothing equations. Then we suggest a new nonmonotone derivative-free line search and combine it into the smoothing Broyden-like method. The proposed algorithm contains the usual monotone line search as a special case and can overcome the difficult of smoothing Newton methods in solving the smooth equations to some extent. Under mild conditions, we prove that the proposed algorithm has global and local superlinear convergence. Furthermore, the algorithm is locally quadratically convergent under suitable assumptions. Preliminary numerical results are also reported.     

A spectral algorithm for large-scale systems of nonlinear monotone equations

A derivative-free iterative scheme that uses the residual vector as search direction for solving large-scale systems of nonlinear monotone equations is presented. It is closely related to two recently proposed spectral residual methods for nonlinear systems which use a nonmonotone line-search globalization strategy and a step-size based on the Barzilai-Borwein choice. The global convergence analysis is presented. In order to study the numerical behavior of the algorithm, it is included an extensive series of numerical experiments. Our computational experiments show that the new algorithm is computationally efficient.     

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.     

A nonmonotone filter trust region method for the system of nonlinear equations

In this paper, we present a nonmonotone filter trust region method to attack the system of nonlinear equations. The system of nonlinear equations is transformed into a constrained nonlinear programming problem at each step: some equations are treated as constraints while the others act as objective functions. Compared with the traditional filter strategies, our algorithm is flexible to accept trail steps by means of the nonmonotone filter technique. Moreover, the restoration phase is not needed so that the scale of the calculation is decreased in a certain degree. Global convergence is proven under some suitable conditions. Numerical experiments also show the efficiency of the algorithm.     

A nonmonotone smoothing Newton method for circular cone programming

The circular cone programming (CCP) problem is to minimize or maximize a linear function over the intersection of an affine space with the Cartesian product of circular cones. In this paper, we study nondegeneracy and strict complementarity for the CCP, and present a nonmonotone smoothing Newton method for solving the CCP. We reformulate the CCP as a second-order cone programming (SOCP) problem using the algebraic relation between the circular cone and the second-order cone. Then based on a one parametric class of smoothing functions for the SOCP, a smoothing Newton method is developed for the CCP by adopting a new nonmonotone line search scheme. Without restrictions regarding its starting point, our algorithm solves one linear system of equations approximately and performs one line search at each iteration. Under mild assumptions, our algorithm is shown to possess global and local quadratic convergence properties. Some preliminary numerical results illustrate that our nonmonotone smoothing Newton method is promising for solving the CCP.     

Preconditioned Spectral Gradient Method

The spectral gradient method is a nonmonotone gradient method for large-scale unconstrained minimization. We strengthen the algorithm by modifications which globalize the method and present strategies to apply preconditioning techniques. The modified algorithm replaces a condition of uniform positive definitness of the preconditioning matrices, with mild conditions on the search directions. The result is a robust algorithm which is effective on very large problems. Encouraging numerical experiments are presented for a variety of standard test problems, for solving nonlinear Poisson-type equations, an also for finding molecular conformations by distance geometry.     

Nonmonotone derivative-free methods for nonlinear equations

In this paper we study nonmonotone globalization techniques, in connection with iterative derivative-free methods for solving a system of nonlinear equations in several variables. First we define and analyze a class of nonmonotone derivative-free linesearch techniques for unconstrained minimization of differentiable functions. Then we introduce a globalization scheme, which combines nonmonotone watchdog rules and nonmonotone linesearches, and we study the application of this scheme to some recent extensions of the Barzilai–Borwein gradient method and to hybrid stabilization algorithms employing linesearches along coordinate directions. Numerical results on a set of standard test problems show that the proposed techniques can be of value in the solution of large-dimensional systems of equations.     

Spectral residual method without gradient information for solving large-scale nonlinear systems of equations

A fully derivative-free spectral residual method for solving large-scale nonlinear systems of equations is presented. It uses in a systematic way the residual vector as a search direction, a spectral steplength that produces a nonmonotone process and a globalization strategy that allows for this nonmonotone behavior. The global convergence analysis of the combined scheme is presented. An extensive set of numerical experiments that indicate that the new combination is competitive and frequently better than well-known Newton-Krylov methods for large-scale problems is also presented.

     

A nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities

In this paper, we investigate a smoothing-type algorithm with a nonmonotone line search for solving a system of equalities and inequalities. We prove that the nonmonotone algorithm is globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are reported.     

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.     

Nonmonotone Inexact Newton Method for the Extended Linear Complementarity Problem

This paper presents a nonmonotone inexact Newton-type method for the extended linear complementarity problem (ELCP). We first reformulate the optimization system of the ELCP problem into a system of smoothed equations. Then we solve this system by a nonmonotone inexact Newton-type algorithm. The global convergence is obtained and numerical tests for some classes of ELCP include linear complementarity, horizontal linear complementarity, and generalized linear complementarity problems are also given to show the e?ciency of the proposed algorithm.     

求解无约束最优化问题的非单调MFR,MPRP方法

本文在求解无约束最优化问题的MFR共轭梯度法和MPRP共轭梯度法中引入两种非单调线性搜索技术.我们证明在适当条件下采用非单调线性搜索的MFR算法和MPRP算法具有全局收敛性.数值结果表明非单调线性搜索具有优越性.     

Gauss–Newton methods with approximate projections for solving constrained nonlinear least squares problems

This paper is concerned with algorithms for solving constrained nonlinear least squares problems. We first propose a local Gauss–Newton method with approximate projections for solving the aforementioned problems and study, by using a general majorant condition, its convergence results, including results on its rate. By combining the latter method and a nonmonotone line search strategy, we then propose a global algorithm and analyze its convergence results. Finally, some preliminary numerical experiments are reported in order to illustrate the advantages of the new schemes.     

Spectral gradient projection method for solving nonlinear monotone equations

An algorithm for solving nonlinear monotone equations is proposed, which combines a modified spectral gradient method and projection method. This method is shown to be globally convergent to a solution of the system if the nonlinear equations to be solved is monotone and Lipschitz continuous. An attractive property of the proposed method is that it can be applied to solving nonsmooth equations. We also give some preliminary numerical results to show the efficiency of the proposed method.     

A nonmonotone approximate sequence algorithm for unconstrained nonlinear optimization

A new nonmonotone algorithm is proposed and analyzed for unconstrained nonlinear optimization. The nonmonotone techniques applied in this algorithm are based on the estimate sequence proposed by Nesterov (Introductory Lectures on Convex Optimization: A Basic Course, 2004) for convex optimization. Under proper assumptions, global convergence of this algorithm is established for minimizing general nonlinear objective function with Lipschitz continuous derivatives. For convex objective function, this algorithm maintains the optimal convergence rate of convex optimization. In numerical experiments, this algorithm is specified by employing safe-guarded nonlinear conjugate gradient search directions. Numerical results show the nonmonotone algorithm performs significantly better than the corresponding monotone algorithm for solving the unconstrained optimization problems in the CUTEr (Bongartz et al. in ACM Trans. Math. Softw. 21:123–160, 1995) library.