A Nonmonotone Trust Region Method for Solving Symmetric Nonlinear Equations

A trust region method combining with nonmonotone technique is proposed tor solving symmetric nonlinear equations. The global convergence of the given method will be established under suitable conditions. Numerical results show that the method is interesting for the given problems.

A New Nonmonotone Adaptive Trust Region Method

The trust region method plays an important role in solving optimization problems. In this paper, we propose a new nonmonotone adaptive trust region method for solving unconstrained optimization problems. Actually, we combine a popular nonmonotone technique with an adaptive trust region algorithm. The new ratio to adjusting the next trust region radius is different from the ratio in the traditional trust region methods. Under some appropriate conditions, we show that the new algorithm has good global convergence and superlinear convergence.

Inexact Newton method via Lanczos decomposed technique for solving box-constrained nonlinear systems

This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the pro- posed algorithm.

Nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization

We propose a nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization. Unlike traditional trust region methods, the subproblem in our method is a simple conic model, where the Hessian of the objective function is approximated by a scalar matrix. The trust region radius is adjusted with a new self-adaptive adjustment strategy which makes use of the information of the previous iteration and current iteration. The new method needs less memory and computational efforts. The global convergence and Q-superlinear convergence of the algorithm are established under the mild conditions. Numerical results on a series of standard test problems are reported to show that the new method is effective and attractive for large scale unconstrained optimization problems.

A New Nonmonotone Trust Region Barzilai-Borwein Method for Unconstrained Optimization Problems

In this paper,we propose a new nonmonotone trust region Barzilai-Borwein(BB for short)method for solving unconstrained optimization problems.The proposed method is given by a novel combination of a modified Metropolis criterion,BB-stepsize and trust region method.The new method uses the reciprocal of BB-stepsize to approximate the Hessian matrix of the objective function in the trust region subproblems,and accepts some bad solutions according to the modified Metropolis criterion based on simulated annealing idea.Under some suitable assumptions,the global convergence of the new method is established.Some preliminary numerical results indicate that,the new method is more efficient compared with the existing trust region BB method.

A NONMONOTONE FILTER LINE SEARCH TECHNIQUE FOR THE MBFGS METHOD IN UNCONSTRAINED OPTIMIZATION

This paper presents a new nonmonotone filter line search technique in association with the MBFGS method for solving unconstrained minimization.The filter method,which is traditionally used for constrained nonlinear programming(NLP),is extended to solve unconstrained NLP by converting the latter to an equality constrained minimization.The nonmonotone idea is employed to the filter method so that the restoration phrase,a common feature of most filter methods,is not needed.The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions.The results of numerical experiments indicate that the proposed method is efficient.

A Nonmonotone Hybrid Method of Conjugate Gradient and Lanczos-type for Solving Nonlinear Systems

In this paper,we construct a new algorithm which combines the conjugate gradient and Lanczos methods for solving nonlinear systems.The iterative direction can be obtained by solving a quadratic model via conjugate gradient and Lanczos methods.Using the backtracking line search,we will find an acceptable trial step size along this direction which makes the objective function nonmonotonically decreasing and makes the norm of the step size monotonically increasing.Global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions.Finally,we present some numerical results to illustrate the effectiveness of the proposed algorithm.

Curvilinear Paths with Nonmonotonic Inexact Line Search Technique for Unconstrained Optimization

In this paper we modify approximate trust region methods via three precon ditional curvilinear paths for unconstrained optimization. To easily form preconditional curvilinear paths within the trust region subproblem, we employ the stable Bunch-Parlett factorization method of symmetric matrices and use the unit lower triangular matrix as a preconditioner of the optimal path and modified gradient path. In order to accelerate the preconditional conjugate gradient path, we use preconditioner to improve the eigenvalue distribution of Hessian matrix. Based on the trial steps produced by the trust region subproblem along the three curvilinear paths providing a direction of sufficient descent, we mix a strategy using both trust region and nonmonotonic line search techniques which switch to back tracking steps when a trial step is unacceptable. Theoretical analysis is given to prove that the proposed algorithms are globally convergent and have a local su-pcrlinear convergent rate under some reasonable conditions. The results of the numerical experiment are reported to show the effectiveness of the proposed algorithms.

CURVILINEAR PATHS AND TRUST REGION METHODS WITH NONMONOTONIC BACK TRACKING TECHNIQUE FOR UNCONSTRAINED OPTIMIZATION

Focuses on a study which examined the modification of type approximate trust region methods via two curvilinear paths for unconstrained optimization. Properties of the curvilinear paths; Description of a method which combines line search technique with an approximate trust region algorithm; Information on the convergence analysis; Details on the numerical experiments.

NONMONOTONIC REDUCED PROJECTED HESSIAN METHOD VIA AN AFFINE SCALING INTERIOR MODIFIED GRADIENT PATH FOR BOUNDED-CONSTRAINED OPTIMIZATION

The authors propose an affine scaling modified gradient path method in association with reduced projective Hessian and nonmonotonic interior backtracking line search techniques for solving the linear equality constrained optimization subject to bounds on variables. By employing the QR decomposition of the constraint matrix and the eigensystem decomposition of reduced projective Hes- sian matrix in the subproblem, the authors form affine scaling modified gradient curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The global convergence and fast local superlinear/quadratical convergence rates of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.

Superlinearly Convergent Affine Scaling Interior Trust-Region Method for Linear Constrained LC^1 Minimization

We extend the classical affine scaling interior trust region algorithm for the linear constrained smooth minimization problem to the nonsmooth case where the gradient of objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear constrained LC1 optimization where the second-order derivative of the objective function is explicitly required to be locally Lipschitzian. The general trust region subproblem in the proposed algorithm is defined by minimizing an augmented affine scaling quadratic model which requires both first and second order information of the objective function subject only to an affine scaling ellipsoidal constraint in a null subspace of the augmented equality constraints. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. Applications of the algorithm to some nonsmooth optimization problems are discussed.

NONMONOTONIC TRUST REGION PROJECTED REDUCED HESSIAN ALGORITHM WITH TWO-PIECE UPDATE FOR CONSTRAINED OPTIMIZATION

This paper proposes a two-piece update of projected reduced Hessian algorithmwith nonmonotonic trust region strategy for solving nonlinear equality constrained optimizationproblems. In order to deal with large problems, a two-piece update of two-side projected reducedHessian is used to replace full Hessian matrix. By adopting the Fletcher's penalty function as themerit function, a nonmonotonic trust region strategy is suggested which does not require the meritfunction to reduce its value in every iteration. The two-piece update of projected reduced Hessianalgorithm which switches to nonmonotonic trust region technique possesses global convergence whilemaintaining a two-step Q-superlinear local convergence rate under some reasonable conditions.Furthermore, one step Q-superlinear local convergence rate can be obtained if at least one of theupdate formulas is updated at each iteration by an alternative update rule. The numerical experimentresults are reported to show the effectiveness of the proposed algorithm.

A New Restarting Adaptive Trust-Region Method for Unconstrained Optimization

In this paper,we present a new adaptive trust-region method for solving nonlinear unconstrained optimization problems.More precisely,a trust-region radius based on a nonmonotone technique uses an approximation of Hessian which is adaptively chosen.We produce a suitable trust-region radius;preserve the global convergence under classical assumptions to the first-order critical points;improve the practical performance of the new algorithm compared to other exiting variants.Moreover,the quadratic convergence rate is established under suitable conditions.Computational results on the CUTEst test collection of unconstrained problems are presented to show the effectiveness of the proposed algorithm compared with some exiting methods.

A NONMONOTONIC TRUST REGION TECHNIQUE FOR NONLINEAR CONSTRAINED OPTIMIZATION

In this paper, a nonmonotonic trust region method for optimization problems with equality constraints is proposed by introducing a nonsmooth merit function and adopting a correction step. It is proved that all accumulation points of the iterates generated by the proposed algorithm are Kuhn-Tucker points and that the algorithm is q-superlinearly convergent.