\(R\)-linear convergence of the Barzilai and Borwein gradient method. (English) Zbl 1002.65069

Dai, Yuhong; Liao, Li-Zhi

IMA J. Numer. Anal. 22, No. 1, 1-10 (2002).

Authors’ abstract: Combined with non-monotone line search, the two-point step size gradient method of J. Barzilai and J. M. Borwein [(BB) ibid. 8, No. 1, 141-148 (1988; Zbl 0638.65055)] has been successfully extended for solving unconstrained optimization problems and is competitive with conjugate gradient methods. In this paper, we establish the \(R\)-linear convergence of the BB method for any-dimensional strongly convex quadratics. One corollary of this result is that the BB method is also locally \(R\)-linear convergent for general objective functions, and hence the stepsize in the BB method will always be accepted by the non-monotone line search when the iterate is close to the solution.

Reviewer: Berwin A.Turlach (Crawley)

MSC:

65K05	Numerical mathematical programming methods
90C25	Convex programming

Keywords:

unconstrained optimization; conjugate gradient methods; \(R\)-linear convergence; strictly convex; two-point step size gradient method; non-monotone line search

Citations:

Zbl 0638.65055