On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds

The authors consider a new algorithm, an interior-reflective Newton approach, 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 using a new affine scaling transformation and following piecewise linear paths (reflection paths). The interior-reflective apporach does not require identification of an ‘activity set’. In this paper the authors establish that the interior-reflective Newton approach is globally and quadratically convergent. Moreover, they develop a specific example of interior-reflective Newton methods which can be used for large-scale and sparse problems.