We consider a family of damped quasi-Newton methods for solving unconstrained optimization problems. This family resembles that of Broyden with line searches, except that the change in gradients is replaced by a certain hybrid vector before updating the current Hessian approximation. This damped technique modifies the Hessian approximations so that they are maintained sufficiently positive defi...

The paper adapts the Newton-like Extremum-Seeking Control technique to extract the maximum power from photovoltaic panels. This technique uses the gradient and Hessian of the panel characteristic in order to approximate the operating point to its optimum. The paper describes in detail the gradient and Hessian estimations carried out by means of sinusoidal dithering signals. Furthermore, we comp...

In this paper, we consider both local and global convergence of the Newton algorithm to solve nonlinear problems when GMRES is used to invert the Jacobian at each Newton iteration. Under weak assumptions, we give a suucient condition for an inexact solution of GMRES to be a descent direction in order to apply a backtracking technique. Moreover, we extend this result to a nite diierence scheme c...

We will present the Newton-Raphson algorithm, and the secant method. A MATLAB function for the Newton-Raphson method. The function.MATLAB has many tools that make this package well suited for numerical computations. The Newton-Raphson method for systems of nonlinear equations.The Newton-Raphson method, or Newton Method, is a powerful technique. Interesting is Section 3, where the birth of the N...

In this paper, we present a modified regularized Newton method for the unconstrained nonconvex optimization by using trust region technique. We show that if the gradient and Hessian of the objective function are Lipschitz continuous, then the modified regularized Newton method (M-RNM) has a global convergence property. Numerical results show that the algorithm is very efficient.

We consider the problem of growth of the sequence of Lebesgue constants corresponding to the Newton interpolation and estimate the growth of this sequence in the case of a nested family of Chebyshev's points.

Newton-Krylov methods, combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with preconditioners destroys in principle the...

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...

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...