On the Convergence of Reflective Newton Methods for Large-scale Nonlinear Minimization Subject to Bounds

Abstract

. We consider a new algorithm, a reflective Newton method, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generates strictly feasible iterates by following piecewise linear paths ("reflection" paths) to generate improved iterates. The reflective Newton approach does not require identification of an "activity set". In this report we establish that the reflective Newton approach is globally and quadratically convergent. Moreover, we develop a specific example of this general reflective path approach suitable for large-scale and sparse problems. 1 Research partially supported by the Applied Mathematical Sciences Research Program (KC04 -02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG0286ER25013. A000, and in part by NSF, AFOSR, and ONR through grant DMS-8920550, and by the Cornell Theory Center which receives major funding from the National Sci...