A Limited-memory Optimization Method Using the Infinitely Many Times Repeated BNS Update and Conjugate Directions

0497050 - ÚI 2020 RIV NL eng J - Journal Article
Vlček, Jan - Lukšan, Ladislav
A Limited-memory Optimization Method Using the Infinitely Many Times Repeated BNS Update and Conjugate Directions.
Journal of Computational and Applied Mathematics. Roč. 351, 1 May (2019), s. 14-28. ISSN 0377-0427. E-ISSN 1879-1778
Institutional support: RVO:67985807
Keywords : Unconstrained minimization * Limited-memory variable metric methods * The repeated Byrd–Nocedal–Schnabel update * The Lyapunov matrix equation * Global convergence * Numerical results
OECD category: Applied mathematics
Impact factor: 2.037, year: 2019
Method of publishing: Limited access
http://dx.doi.org/10.1016/j.cam.2018.10.054

To improve the performance of the limited-memory variable metric L-BFGS method for large scale unconstrained optimization, repeating of some BFGS updates was proposed e.g. in Al-Baali (1999, 2002). Since the repeating process can be time consuming, the suitable extra updates need to be selected carefully. We show that for the limited-memory variable metric BNS method, matrix updating can be efficiently repeated infinitely many times under some conditions, with only a small increase of the number of arithmetic operations. The limit matrix can be written as a block BFGS update (Vlček and Lukšan, 2018), which can be obtained by solving of some low-order Lyapunov matrix equation. The resulting method can be advantageously combined with methods based on vector corrections for conjugacy, see e.g. Vlček and Lukšan (2015). Global convergence of the proposed algorithm is established for convex and sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new method.
Permanent Link: http://hdl.handle.net/11104/0289654