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A family of spectral gradient methods for optimization
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abstract
We propose a family of spectral gradient methods, whose stepsize is determined by a convex combination of the long Barzilai-Borwein (BB) stepsize and the short BB stepsize. Each member of the family is shown to share certain quasi-Newton property in the sense of least squares. The family also includes some other gradient methods as its special cases. We prove that the family of methods is $R$-superlinearly convergent for two-dimensional strictly convex quadratics. Moreover, the family is $R$-linearly convergent in the any-dimensional case. Numerical results of the family with different settings are presented, which demonstrate that the proposed family is promising.
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Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension $n\geq 4$
For every dimension n≥4, open families of strictly convex quadratics and starting points exist where BB1 converges no faster than a fixed geometric rate, ruling out superlinear convergence on a positive-measure set.
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