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The study presents the modification of the Broyden-Flecher-Goldfarb-Shanno (BFGS) update (H-Version) based on the determinant property of inverse of Hessian matrix (second derivative of the objective function), via updating of the vector s ( the difference between the next solution and the current solution), such that the determinant of the next inverse of Hessian matrix is equal to the determinant of the current inverse of Hessian matrix at every iteration. Moreover, the sequence of inverse of Hessian matrix generated by the method would never approach a near-singular matrix, such that the program would never break before the minimum value of the objective function is obtained. Moreover, the new modification of BFGS update (H-version) preserves the symmetric property and the positive definite property without any condition.
The focus of this article is to add a new class of rank one of modified Quasi-Newton techniques to solve the problem of unconstrained optimization by updating the inverse Hessian matrix with an update of rank 1, where a diagonal matrix is the first component of the next inverse Hessian approximation, The inverse Hessian matrix is generated by the method proposed which is symmetric and it satisfies the condition of modified quasi-Newton, so the global convergence is retained. In addition, it is positive definite that guarantees the existence of the minimizer at every iteration of the objective function. We use the program MATLAB to solve an algorithm function to introduce the feasibility of the proposed procedure. Various numerical examples are given`.
The aim of this paper is to introduce the new class of Broyden convex update by using modified symmetric rank1 update together with modified BFGS update for inverse of hessian matrix where the parameter in used is Hoshino parameter, numerical examples are solved and introduce the convergence of the proposed class and the results are reported as a table of computation.
The objective of this article is to solve the unconstrained optimization problem by using Marquardt method together with MQ-N (modified quasi-Newton) method. The Hessian matrix will be computed numerically by using modified Broyden-Flechert-Goldfarb-Shanno (BFGS) update ( H-version) after convert it to the B-version by using Sherman-Morisson-Woodburge (SMW) formula which guarantee the two important properties SPD (symmetric and positive definite), and hence the exact second derivative of OF (objective function) does not be needed to compute. The line search technique is very important to accelerate the method to terminate at the minimum value of OF, so in this article the line search technique is instead by used the search technique of Marquardt method together with MQ-N method (especially modified BFGS method where the step size is equal one) to solve the unconstrained optimization problem. Test problems are solved by Matlab software to prove the effective of this new technique so called the Marquardt extended technique (MET).
In this research we introduced a new update of the Hessian matrix or we updating only the diagonal elements of Hessian matrix, and make the non-diagonal elements always equal to zero and in this case we can preserve the sparse property so called the Diagonal Update.
In this work, we propose modification for PSB update with a new extended Quasi – Newton condition for unconstrained optimization problem, so it’s called (α – PSB) method. PSB update kind of rank two update, which solve the unconstrained optimization problem, but this update can’t guarantee the positive definite property of Hessian matrix. In this work, the guarantee of positive definite property for the Hessian matrix be confirmed by updating the vector sk, which represent the difference between the next gradient and the current gradient of the objective function which assume to be continuous twice and differentiable. Then we proved the existentialism of this update. Numerical results are reported where the comparison between the proposed method and the PSB update method under standard problems.
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