A Modified Globally Convergent Self-Scaling BFGS Algorithm for Unconstrained Optimization

Abstract

In this paper, a modified globally convergent self-scaling BFGS algorithm for solving convex unconstrained optimization problems was investigated in which it employs exact line search strategy and the inverse Hessian matrix approximations were positive definite. Experimental results indicate that the new proposed algorithm was more efficient than the standard BFGS-algorithm.

New Hybrid (Oren- Al-Bayati) Self-Scaling Algorithm With Armijo Line-Search

“…Al-Bayati [1] found another interesting family of VM -updates of (1) by further scaling of Oren's family of updates with a scalar  k  such that:…”

“…Al-Bayati [1] has a search direction which is identical to the standard CG-direction (see the following theorem):…”

A New Globally Convergent Self-Scaling Vm Algorithm for Convex and Nonconvex Optimization

“…To prove that the new updates ) 25 ( generate identical conjugate gradient search directions if the function is quadratic and the exact line searches are used, let us consider the following property: Property (3.1.1): Let f be given by Proof: see (Al-Bayati, 1991). Powell (1983) showed that if the level set  …”

“…Li and Fukushima proposed a modified BFGS methods based on a new Quasi -Newton equation x (Storey & HU., 1993). One then computes the next iteration by the formula (Al-Bayati, 1991) modified the BFGS formula by writing:…”

A new self-scaling VM-algorithm for non-convex optimization, part 1

“…We study the global convergence of a self -scaling Al-Bayati [4], VM-method with non-monotone line searches for solving the unconstrained optimization problem (see [3]and [12])…”