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The cyclic Barzilai-Borwein method for unconstrained optimization. (English) Zbl 1147.65315
Summary: In the cyclic Barzilai-Borwein (CBB) method, the same Barzilai-Borwein (BB) stepsize is reused for $m$ consecutive iterations. It is proved that CBB is locally linearly convergent at a local minimizer with positive definite Hessian. Numerical evidence indicates that when $m>n/2\ge 3$, where $n$ is the problem dimension, CBB is locally superlinearly convergent. In the special case $m=3$ and $n=2$, it is proved that the convergence rate is no better than linear, in general. An implementation of the CBB method, called adaptive cyclic Barzilai-Borwein (ACBB), combines a non-monotone line search and an adaptive choice for the cycle length $m$. In numerical experiments using the CUTEr test problem library, ACBB performs better than the existing BB gradient algorithm, while it is competitive with the well-known PRP+ conjugate gradient algorithm.
##### MSC:
 65K05 Mathematical programming (numerical methods) 90C20 Quadratic programming 90C25 Convex programming 90C30 Nonlinear programming
##### Software:
CUTEr; CG_DESCENT