Two limited-memory optimization methods with minimum violation of the previous secant conditions

• Published in: Computational optimization and applications Vol. 80; no. 3; pp. 755 - 780
• Main Authors: Vlček, Jan, Lukšan, Ladislav
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2021; Springer Nature B.V
• Subjects: Algorithms; Computer science; Convergence; Convex and Discrete Geometry; Guarantees; Management Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimization; Statistics; Global convergence; Variable metric methods; Variationally derived methods; 65F30; Unconstrained minimization; 65K10; Numerical results; Limited-memory methods
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-021-00318-y

Limited-memory variable metric methods based on the well-known Broyden-Fletcher-Goldfarb-Shanno (BFGS) update are widely used for large scale optimization. The block version of this update, derived for general objective functions in Vlček and Lukšan (Numerical Algorithms 2019), satisfies the secant conditions with all used difference vectors and for quadratic objective functions gives the best improvement of convergence in some sense, but the corresponding direction vectors are not descent directions generally. To guarantee the descent property of direction vectors and simultaneously violate the secant conditions as little as possible in some sense, two methods based on the block BFGS update are proposed. They can be advantageously used together with methods based on vector corrections for conjugacy. Here we combine two types of these corrections to satisfy the secant conditions with both the corrected and uncorrected (original) latest difference vectors. Global convergence of the proposed algorithm is established for convex and sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new methods.