A block version of the BFGS variable metric update formula is investigated. It satisfies the quasi-Newton conditions with all used difference vectors and gives the best improvement of convergence in some sense for quadratic objective functions, but it does not guarantee that the direction vectors are descent for general functions. To overcome this difficulty and utilize the advantageous properties of the block BFGS update, a block version of the limited-memory BNS method for large scale unconstrained optimization is proposed. The algorithm is globally convergent for convex sufficiently smooth functions and our numerical experiments indicate its efficiency.
@article{703010,
title = {A generalized limited-memory BNS method based on the block BFGS update},
booktitle = {Programs and Algorithms of Numerical Mathematics},
series = {GDML\_Books},
publisher = {Institute of Mathematics CAS},
address = {Prague},
year = {2017},
pages = {164-171},
url = {http://dml.mathdoc.fr/item/703010}
}
Vlček, Jan; Lukšan, Ladislav. A generalized limited-memory BNS method based on the block BFGS update, dans Programs and Algorithms of Numerical Mathematics, GDML_Books, (2017), pp. 164-171. http://gdmltest.u-ga.fr/item/703010/