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Alternate step gradient method. (English) Zbl 1056.65055
Motivated by how to avoid producing zigzags, the author proposes the alternate step (AS) gradient method. The AS method uses an steepest descent step and a J. Barzilai and J. M. Borwein (BB) [IMA J. Numer. Anal. 8, No. 1, 141–148 (1988; Zbl 0638.65055)] step alternately, and is superior to the BB method for symmetric and positive definite linear systems. $$R$$-linear convergence is established for a collection of gradient methods including the AS method. Gradient methods related to the AS method and an extension of the AS method to unconstrained optimization are also discussed. A more efficient gradient algorithm for unconstrained optimization is also provided. Some interesting insights into gradient methods and discussion about monotonicity and nonmonotonicity are also given.

##### MSC:
 65K05 Numerical mathematical programming methods 90C52 Methods of reduced gradient type
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