## 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 $$m>n/2\geq 3$$, 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 Numerical mathematical programming methods 90C20 Quadratic programming 90C25 Convex programming 90C30 Nonlinear programming

### Software:

CUTEr; CG_DESCENT
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