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It is well known that search directions in nonlinear conjugate gradient (CG)can sometimes become nearly dependent, causing a dramatic slow-down in theconvergence rate. We provide a theoretical analysis of this loss ofindependence. The analysis applies to the case of a strictly convex objectivefunction and is motivated by older work of Nemirovsky and Yudin. Loss ofindependence can affect several of the well-known variants of nonlinear CGincluding Fletcher-Reeves, Polak-Ribi\`ere (nonnegative variant), andHager-Zhang.  Based on our analysis, we propose a relatively inexpensive computational testfor detecting loss of independence. We also propose a method for correcting itwhen it is detected, which we call "subspace optimization." Although thecorrection method is somewhat expensive, our experiments show that in somecases, usually the most ill-conditioned ones, it yields a method much fasterthan any of these three variants. Even though our theory covers only stronglyconvex objective functions, we provide computational results to indicate thatthe detection and correction mechanisms may also hold promise for nonconvexoptimization.

Detecting and correcting the loss of independence in nonlinear conjugate gradient