A globally convergent BFGS method with nonmonotone line search for non-convex minimization

Abstract

a b s t r a c tIn this paper, we propose a modified BFGS (Broyden-Fletcher-Goldfarb-Shanno) method with nonmonotone line search for unconstrained optimization. Under some mild conditions, we show that the method is globally convergent without a convexity assumption on the objective function. We also report some preliminary numerical results to show the efficiency of the proposed method.Crown

Blind Source Separation of Continuous-Time Chaotic Signals Based on Fast Random Search Algorithm

“…Proof: From the very definition of the optimization scheme in [11] and [12], there exists an auxiliary parameter vectorq on D, and, clearly, max{U (q), k ≤ v} → U (q * ) as v → ∞. The proof now follows since U (q) ≤ U (q k ).…”

“…An important feature of the MBFGS is that the function value at each iteration allows for an occasional decrease. Compared with some extant method in [12], the MBFGS can converge to a local optimal point without a convex assumption on the objective function. Additionally, the MBFGS can be considered as an extension of the method in [13] to the nonmonotone scheme.…”

“…The basic idea of the FRS algorithm is to utilize the modified Broyden-Fletcher-Goldfarb-Shanno (MBFGS) method [12] with nonmonotone line search to replace the random sampling in the ARS. The MBFGS is an unconstrained optimization method with global and extremely fast convergence.…”

A nonmonotone modified BFGS algorithm for nonconvex unconstrained optimization problems

“…Their modifications were so useful that have motivated many researchers to make further improvements on the BFGS method. For example, Xiao et al introduced a new algorithm by using the MBFGS update formula suggested by Li and Fukushima along with a nonmonotone line search proposed in [23]. They proved that the method is globally convergent for nonconvex optimization problems.…”

“…In this paper, a nonmonotone MBFGS algorithm is introduced and the global convergence of the method is proved without convexity assumption. Actually, the algorithm combines the MBFGS method, proposed by Xiao et al in [23], with nonmonotone line search (5) and also gains advantages of [2] and [24]. Numerical experiments indicate that the new algorithm is promising and efficient.…”