Convergence Properties of a Modified BFGS Algorithm for Minimization with Armijo-Goldstein Steplengths

Abstract

The line search strategy is crucial for an efficient unconstrained optimization algorithm. One of the reason why the Wolfe line searches is recommended lies in that it ensures positive definiteness of BFGS updates. When gradient information has to be obtained costly, the Armijo-Goldstein line searches may be preferred. To maintain positive definiteness of BFGS updates based on the Armijo-Goldstein line searches, a slightly modified form of BFGS update is proposed by I.D. Coope and C.J. Price (Journal of Computational Mathematics, 13 (1995), 156-160), while its convergence properties is open up to now. This paper shows that the modified BFGS algorithm is globally and superlinearly convergent based on the Armijo-Goldstein line searches.