Iranian Journal of Operations Research Vol. 3, No. 1, 2012, pp. 1-10 On the B ehavior of Damped Quasi-Newton Metho ds for U nconstrained Optimization M. Al-Baali 1,* , L. Grandinetti 2 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 definite. Hence, the objective function is reduced sufficiently on each iteration. The recent result that the damped technique maintains the global and superlinear convergence properties of a restricted class of quasi-Newton methods for convex functions is tested on a set of standard unconstrained optimization problems. The behavior of the methods is studied on the basis of the numerical results required to solve these test problems. It is shown that the damped technique improves the performance of quasi-Newton methods substantially in some robust cases (as the BFGS method) and significantly in certain inefficient cases (as the DFP method). Keywords: Unconstrained optimization, Quasi-Newton methods, Line-search framework. Manuscript received on 22/10/2011 and accepted for publication on 31/12/2011.g 1. Introduction We study the behavior of the recent class of damped quasi-Newton methods, proposed by Al-Baali [7] for solving the unconstrained optimization problem ( ) min , x R f x ∈ where f is a nonlinear differentiable function. This damped (D-) class resembles that of Broyden with line searches (see, for example, Dennis and Schnab el [11], Fletcher [12] or Nocedal and Wright [26]) except that the change in gradients 1 k k k g g γ + = − is replaced by the hybrid damp ed-technique k γ  = ( ) 1 , k k k k k B ϕ γ ϕ δ + − (1) where ( ] 0,1 k ϕ ∈ is a parameter, before updating a Hessian approximation k B . Here, ( ) ( ) 2 1 , , k k k k k k k g f x B f x x x δ + =∇ ≈∇ = − and k x is the current estimate of a solution of the problem. _________________________ gInvited paper. *Corresponding Author. 1 Department of Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman. Email: albaali@squ.edu.om. 2 Department of Electronics, Informatics and Systems, Calabria University,Rende 87036, Italy. Email: lugran@unical.it. Downloaded from iors.ir at 17:34 +0430 on Thursday July 27th 2017