In this paper, we solve unconstrained optimization problem using a free line search steepest descent method. First, we propose a double parameter scaled quasi Newton formula for calculating an approximation of the Hessian matrix. The approximation obtained from this formula is a positive definite matrix that is satisfied in the standard secant relation. We also show that the largest eigen value...

In this study we proposed two Quasi-Newton methods to deal with traffic assignment in the capacitated network. The methods combine Newton formula, column generation and penalty techniques. The first method employ the gradient of the objective function to obtain an improving feasible direction scaled by the second-order derivatives. The second one is to employ Rosen gradient to obtain an improvi...

The conjugate gradient method plays an important role in solving large-scaled problems and the quasi-Newton method is known as the most efficient method in solving unconstrained optimization problems. Therefore, in this paper, the new hybrid 2592 Mohd Asrul Hery Ibrahim et al. method between the conjugate gradient method and the quasi-newton method for solving optimization problem is suggested....

We consider proximal gradient methods for minimizing a composite function of differentiable and convex function. To accelerate the general methods, we focus on quasi-Newton type based mappings scaled by matrices. Although it is usually difficult to compute mappings, applying memoryless symmetric rank-one (SR1) formula makes this easier. Since (quasi-Newton) matrices must be positive definite, d...

This paper concerns the memoryless quasi-Newton method, that is precisely the quasi-Newton method for which the approximation to the inverse of Hessian, at each step, is updated from the identity matrix. Hence its search direction can be computed without the storage of matrices. In this paper, a scaled memoryless symmetric rank one (SR1) method for solving large-scale unconstrained optimization...

We provide a formula for variational quasi-Newton updates with multiple weighted secant equations. The derivation of the formula leads to a Sylvester equation in the correction matrix. Examples are given.

<p style='text-indent:20px;'>Memoryless quasi–Newton updating formulas of BFGS (Broyden–Fletcher–Goldfarb–Shanno) and DFP (Davidon–Fletcher–Powell) are scaled using well-structured diagonal matrices. In the scaling approach, elements as well eigenvalues memoryless play significant roles. Convergence analysis given diagonally methods is discussed. At last, performance numerically tested on...

In this paper, we consider the problem of efficiently computing the eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact formulation. In addition, we produce a compact formula for quasi-Newton matrices generated by any member of the Broyden convex class of updates. Our proposed method makes use of efficient updates to the QR factorization that substantially reduces the cos...

A new family of limited-memory variable metric or quasi-Newton methods for unconstrained minimization is given. The methods are based on a positive definite inverse Hessian approximation in the form of the sum of identity matrix and two low rank matrices, obtained by the standard scaled Broyden class update. To reduce the rank of matrices, various projections are used. Numerical experience is e...

In this paper, we investigate the semismooth Newton and quasi-Newton methods for the minimization problem in the weighted `−regularization of nonlinear inverse problems. We propose the conditions for obtaining the convergence of two methods. The semismooth Newton method is proven to locally converge with superlinear rate and the semismooth quasi-Newton method is proven to locally converge at le...