Convergence Properties of a Modified BFGS Algorithm for Minimization with Armijo-Goldstein Steplengths (Journal of Computational Mathematics, 2021)
The line search strategy is crucial for an efficient unconstrained optimizationalgorithm. One of the reason why the Wolfe line searches is recommended lies inthat it ensures positive definiteness of BFGS updates. When gradient informationhas to be obtained costly, the Armijo-Goldstein line searches may be preferred. Tomaintain positive definiteness of BFGS updates based on the Armijo-Goldstein linesearches, a slightly modified form of BFGS update is proposed by I.D. Coope andC.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.