An augmented memoryless BFGS method based on a modified secant equation with application to compressed sensing

Abstract

Making a rank-one modification on the classical BFGS (Broyden–Fletcher–Goldfarb–Shanno) updating formula, we develop a class of augmented BFGS methods. The suggested formula can be considered as a hybridization of the basic BFGS updating formula for Hessian with an additional rank-one term embedded to guarantee a general modified secant equation. By using the well-known Sherman–Morrison formula, the inverse of a memoryless version of the given updating formula is computed to be applied for solving large-scale problems. Convergence analysis is concisely carried out as well. At last, the practical merits of the method are investigated by numerical tests on a set of CUTEr problems as well as the well-known compressed sensing problem. Results show the computational efficiency of the given method.