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The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. (English) Zbl 0898.90119

The Barzilai-Borwein method [J. Barzilai and J. M. Borwein, IMA J. Numer. Anal. 8, No. 1, 141-148 (1988; Zbl 0638.65055)] requires few storage locations and very inPRRexpensive computations. Since a special step length is used no line search is required. A descent in the objective function is not guaranteed in every step. However, the global convergence has been established for the convex quadratic case. For the nonquadratic case the method needs to be incorporated in a globalization scheme.

In this work the author proposes the combination of a nonmonotone line search strategy with the Barzilai-Borwein method (Global Barzilai-Borwein GBB).

Thus the global convergence is guaranteed. For 15 test problems the new method is compared with two implementations of the conjugate gradient method (routine CONMIN of Shanno and Phua; Polak-Ribiere implementation ${\text{PR}}^{+}$ of Gilbert and Nocedal). As was expected CONMIN and ${\text{PR}}^{+}$ out perform the GBB in a number of iterations except for problems with a well-conditioned Hessian at the solution, in which case the number of iterations is similar.

##### MSC:
 90C30 Nonlinear programming 90C52 Methods of reduced gradient type 90C06 Large-scale problems (mathematical programming)
CONMIN