The method of shortest residuals for large scale nonlinear problems

The paper discusses the method of shortest residuals for nonlinear programming problems. In [R. Pytlak, IMA J. Numer. Anal., 14 (1994), pp. 443-460] we presented a family of conjugate gradient algorithms which originated in the method of the shortest residuals and which has a strong resemblance with conjugate gradient algorithms by Lemarechal and Wolfe. We proved global convergence of Polak-Ribiere version of the method. The method of shortest residuals was further analysed by Dai and Yuan [Numerische Mathematik, 83 (1999), pp. 581-598]. In this paper we show that our Fletcher-Reeves version, which does not require restarts, is also globally convergent. Furthermore, we show sufficiency conditions for the Fletcher-Reeves version to be globally convergent for problems with box constraints. Finally, we provide results of our numerical experiments with several versions of the method of shortest residuals.