A limited-memory optimization method using the infinitely many times repeated BNS update and conjugate directions
A new limited-memory variable metric method for large-scale unconstrained optimization is proposed. After a brief description of the L-BFGS and BNS methods, the authors derive and describe the properties of the BNS matrix update for the limited-memory variable metric BNS method for minimization of general functions. This update can be efficiently repeated infinitely many times under some conditions and without a noticeable increase of the number of arithmetic operations. The resulting limit update is a block BFGS update described in [\textit{J. Vlček} and \textit{L. Lukšan}, Numer. Algorithms 80, No. 3, 957--987 (2019; Zbl 1440.90076)], is more efficient than the L-BFGS update, and can be obtained by solving of a suitable Lyapunov matrix equation. The order of this equation can be decreased by combination with methods based on vector corrections for conjugacy. An efficient method for solving the corresponding low-order Lyapunov equations numerically is also proposed. Global convergence of the presented algorithm is established for convex and sufficiently smooth functions. The efficiency of the algorithm is reported using extensive numerical experiments performed on three collections of test problems. These experiments also indicate that the new approach can improve unconstrained large-scale minimization results significantly compared with the frequently used L-BFGS method. The paper is quite technical. Only the reader having some experience in the field and being familiar with the BFGS and BNS methods will understand the paper well.
scientific article; zbMATH DE number 7007554
scientific article