The authors present the multivariate spectral gradient (MSG) method for solving unconstrained optimization problems. Combined with some quasi-Newton property the MSG method allows an individual adaptive stepsize along each coordinate direction, which guarantees that the method is finitely convergent for positive definite quadratics. Especially, it converges no more than two steps for positive definite quadratics with diagonal Hessian, and quadratically for objective functions with positive definite diagonal Hessian. Moreover, based on a nonmonotone line search, global convergence is established for the MSG algorithm.
Also a numerical study of the MSG algorithm compared with the global Barzilai-Borwein (GBB) algorithm is given. The search direction of the MSG method is close to that presented in the paper by M. N. Vrahatis, G. S. Androulakis, J. N. Lambrinos and G. D. Magoulas [J. Comput. App. Math. 114, 367–386 (2000; Zbl 0958.65072)], but the explanation for the steplength selection is different. The stepsize in this method is selected from the estimates of the eigenvalues of the Hessian but not a local estimation of the Lipschitz constant in the above mentioned paper. At last numerical results are reported, which show that this method is promising and deserves futher discussing.