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On a Faster $R$-Linear Convergence Rate of the Barzilai-Borwein Method
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abstract
The Barzilai-Borwein (BB) method has demonstrated great empirical success in nonlinear optimization. However, the convergence speed of BB method is not well understood, as the known convergence rate of BB method for quadratic problems is much worse than the steepest descent (SD) method. Therefore, there is a large discrepancy between theory and practice. To shrink this gap, we prove that the BB method converges $R$-linearly at a rate of $1-1/\kappa$, where $\kappa$ is the condition number, for strongly convex quadratic problems. In addition, an example with the theoretical rate of convergence is constructed, indicating the tightness of our bound.
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Kahan's Automatic Step-Size Control for Unconstrained Optimization
Kahan's KGD step-size is shown to converge at least R-linearly with rate 1-1/cond(H) for quadratics, and an adaptive generalization for general optimization is proved and tested.
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