This short note considers Hessian updates for a sequential quadratically constrained quadratic programming (SQCQP) method. The SQCQP method is an iterative method to solve an inequality constrained optimization problem. At each iteration, the SQCQP method solves a quadratically constrained quadratic programming subproblem whose objective function and constraints are approximations to the object...

Aiming at the problems of low reconstruction rate and poor precision when reconstructing sparse signals in wireless sensor networks, a signal algorithm based on Limit-Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) quasi-Newton method is proposed. The L-BFGS uses two-loop recursion to find descent direction dk directly by calculating step difference between m adjacent iteration points, matrix Hk appr...

Limited-memory BFGS quasi-Newton methods approximate the Hessian matrix of second derivatives by the sum of a diagonal matrix and a fixed number of rank-one matrices. These methods are particularly effective for large problems in which the approximate Hessian cannot be stored explicitly. It can be shown that the conventional BFGS method accumulates approximate curvature in a sequence of expandi...

A new method for quasi-Newton minimization outperforms BFGS by combining least-change updates of the Hessian with step sizes estimated from a Wishart model of uncertainty. The Hessian update is in the Broyden family but uses a negative parameter, outside the convex range, that is usually regarded as the safe zone for Broyden updates. Although full Newton steps based on this update tend to be to...

This paper tackles the problem of computing the Karcher mean of a collection of symmetric positive-definite matrices. We present a concrete limited-memory Riemannian BFGS method to handle this computational task. We also provide methods to produce efficient numerical representations of geometric objects on the manifold of symmetric positive-definite matrices that are required for Riemannian opt...

Linear minimum mean square error (MMSE) detection achieves a good trade-off between performance and complexity for massive multiple-input multiple-output (MIMO) systems. To avoid the high-dimensional matrix inversion involved, MMSE can be transformed into an unconstrained optimization problem then solved by efficient numerical algorithms in iterative way. Three low-complexity Broyden-Fletcher-G...

This work shows that the BFGS method and other methods in the Broyden class, with exact line searches, may fail for non-convex objective functions.

We investigate the BFGS algorithm with an inexact line search when applied to nonsmooth functions, not necessarily convex. We define a suitable line search and show that it generates a sequence of nested intervals containing points satisfying the Armijo and weak Wolfe conditions, assuming only absolute continuity. We also prove that the line search terminates for all semi-algebraic functions. T...