Recently, Fan [4, Math. Comput., 81 (2012), pp. 447-466] proposed a modified Levenberg-Marquardt (MLM) method for nonlinear equations. Using a trust region technique, global and cubic convergence of the MLM method is proved [4] under the local error bound condition, which is weaker than nonsingularity. The purpose of the paper is to investigate the convergence properties of the MLM method with ...

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 uncon-strained minimization of differentiable functions. Then we introduce a globalization scheme, which combines nonmonoton...

When solving inequality constrained optimization problems via Sequential Quadratic Programming (SQP), it is potentially advantageous to generate iterates that all satisfy the constraints: all quadratic programs encountered are then feasible and there is no need for a surrogate merit function. (Feasibility of the successive iterates is in fact required in many contexts such as in real-time appli...

In this paper, we present a nonmonotone adaptive trust region method for unconstrained optimization based on conic model. The new method combines nonmonotone technique and a new way to determine trust region radius at each iteration. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments show that our algorithm is effective.

The work by Gould, Loh, and Robinson [A filter method with unified step computation for nonlinear optimization, SIAM J. Optim., 24 (2014), pp. 175–209 ] established global convergence of a new filter line search method for finding local first-order solutions to nonlinear and nonconvex constrained optimization problems. A key contribution of that work was that the search direction was computed u...

In this paper, we incorporate a nonmonotone technique with the new proposed adaptive trust region radius (Shi and Guo, 2008) [4] in order to propose a new nonmonotone trust region method with an adaptive radius for unconstrained optimization. Both the nonmonotone techniques and adaptive trust region radius strategies can improve the trust region methods in the sense of global convergence. The g...

We propose a new truncated Newton method for large scale unconstrained optimization, where a Conjugate Gradient (CG)-based technique is adopted to solve Newton’s equation. In the current iteration, the Krylov method computes a pair of search directions: the first approximates the Newton step of the quadratic convex model, while the second is a suitable negative curvature direction. A test based...

The (almost) 1-cover lifting property of omega-limit sets is established for nonmonotone skew-product semiflows, which are comparable to uniformly stable eventually strongly monotone skew-product semiflows. These results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of ODEs, reaction-diffusion systems, differential systems with time delays ...

We propose a new truncated Newton method for large scale unconstrained optimization, where a Conjugate Gradient (CG)-based technique is adopted to solve Newton’s equation. In the current iteration, the Krylov method computes a pair of search directions: the first approximates the Newton step of the quadratic convex model, while the second is a suitable negative curvature direction. A test based...