Modified BFGS Update (H-Version) Based on the Determinant Property of Inverse of Hessian Matrix for Unconstrained Optimization

Abstract

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.

New Class of Rank 1 Update for Solving Unconstrained Optimization Problem

“…A new family of updated BFGS updates was proposed in [9] to resolve the unconstrained issue of optimization for non-convex functions based on a new modified weak Wolfe-Powell line search technique. In [10], the authors proposed the modified BFGS update (H-version) by updating the vector s ( next solution-current solution) and they provided that the proposed method preserve the strong positive definite property and globally convergent. The method of Newton ( ( ( is efficient because it uses the Hessian matrix that provides the useful curvature data, however the computational efforts of the Hessian matrix are very costly or it is difficult to test the Hessian matrix even though the Hessian matrix is not analytically usable this refers to a class of techniques that use only the values of the equation and the gradients of the objective function which are closely related to the method of Newton.…”

“…( 1 ) The objective function f is assumed to be twice continuous and differentiable in a convex open set D, moreover f is uniformly convex.The most efficient quasi-Newton method is the (BFGS( method, which is introduced independently by Broyden, Fletcher, Goldfarb, and Shannoin in [10]. Note that the Hessian approximation B k+1 can be modified in the BFGS approach and ( ) ( )…”

Modified Broyden Convex Class Update with Hoshino parameter

“…and in [6] introduce the modified quasi-Newton for Modified Broyden Fletcher Goldfarb shanno such that :…”

“…Since the Broyden Fletcher Goldfarb shanno (BFGS) is positive definite if and only if 𝕤 k 𝕪 k T > 0 there for in [6] interdicted Modified Broyden Fletcher Goldfarb shanno (MBFGS) update which is positive definite without conditions and the (Hversion) of this method from the source [6] is :…”

Marquardt Extended Technique for Solving Unconstrained Optimization Problems

“…[6] modified the Powell Symmetric Broyden (MPSB ) update to guarantee the PD property of Hessian matrix where the original update does not guarantee this property. [7] Proposed the MBFGS update by updating the vector s (next solution-current solution) and provided the preserve of the strong PD property and globally convergent. This article introduces a new methodology for solving the unconstrained optimization issue by combining the Marquardt method and the MQ-N method.…”