A Limited Memory Gradient Projection Method for Box-Constrained Quadratic Optimization Problems

Gradient Projection (GP) methods are a very popular tool to address box-constrained quadratic problems thanks to their simple implementation and low computational cost per iteration with respect, for example, to Newton approaches. It is however possible to include, in GP schemes, some second order information about the problem by means of a clever choice of the steplength parameter which controls the decrease along the anti-gradient direction. Borrowing the analysis developed by Barzilai and Borwein (BB) for an unconstrained quadratic programming problem, in 2012 Roger Fletcher proposed a limited memory steepest descent (LMSD) method able to effectively sweep the spectrum of the Hessian matrix of the quadratic function to optimize. In this work we analyze how to extend the Fletcher’s steplength selection rule to GP methods employed to solve box-constrained quadratic problems. Particularly, we suggest a way to take into account the lower and the upper bounds in the steplength definition, providing also a theoretical and numerical evaluation of our approach.