Computational Optimization and Applications
A new gradient method via quasi-Cauchy relation which guarantees descent
Journal of Computational and Applied Mathematics
Hybrid spectral gradient method for the unconstrained minimization problem
Journal of Global Optimization
Iterative regularization algorithms for constrained image deblurring on graphics processors
Journal of Global Optimization
Journal of Computational and Applied Mathematics
The Chaotic Nature of Faster Gradient Descent Methods
Journal of Scientific Computing
Modified subspace Barzilai-Borwein gradient method for non-negative matrix factorization
Computational Optimization and Applications
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The Barzilai-Borwein (BB) gradient method, and some other new gradient methods have shown themselves to be competitive with conjugate gradient methods for solving large dimension nonlinear unconstrained optimization problems. Little is known about the asymptotic behaviour, even when applied to n−dimensional quadratic functions, except in the case that n=2. We show in the quadratic case how it is possible to compute this asymptotic behaviour, and observe that as n increases there is a transition from superlinear to linear convergence at some value of n≥4, depending on the method. By neglecting certain terms in the recurrence relations we define simplified versions of the methods, which are able to predict this transition. The simplified methods also predict that for larger values of n, the eigencomponents of the gradient vectors converge in modulus to a common value, which is a similar to a property observed to hold in the real methods. Some unusual and interesting recurrence relations are analysed in the course of the study.