The global and superlinear convergence of a new nonmonotone MBFGS algorithm on convex objective functions

Abstract

In this paper, a new nonmonotone MBFGS algorithm for unconstrained optimization will be proposed. Under some suitable assumptions, the global and superlinear convergence of the new nonmonotone MBFGS algorithm on convex objective functions will be established. Some numerical experiments show that this new nonmonotone MBFGS algorithm is competitive to the MBFGS algorithm and the nonmonotone BFGS algorithm.

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

“…We also tested our method on a large number of problems from CUTEr library, which indicates our method is promising. Lately, [16] has already proven the superlinear convergence when using Wolfe condition. Although we all suppose that superlinear convergence holds as for the Armijo line search condition, at present we cannot give a complete proof of it yet.…”

“…Their numerical experiments showed that this method was competitive to the standard BFGS algorithm. More recently, Liu, Yao, and Wei (see [16]) also introduced a modified nonmonotone BFGS algorithm on the basis of a modified secant condition. Unfortunately, the global convergence result of this method depended on the convex assumption of the objective functions.…”

“…We will propose a modified nonmonotone BFGS method that is globally convergent without a convex assumption on the objective functions. We will show that the global convergence is achieved even if the Armijo line search is used, while in [12,15,16], the Wolfe conditions were used; in [26], a modified Armijo line search was used. The numerical experiments on a set of problems from CUTEr collection indicate that the proposed method is practically effective.…”

The convergence of a new modified BFGS method without line searches for unconstrained optimization or complexity systems

“…Proof From the proof process of the Lemma 3.1 in [17], we can get the result. Lemma 3 Let the sequence {x k } be generated by Algorithm 1.…”

A nonmonotone modified BFGS algorithm for nonconvex unconstrained optimization problems

“…The nonmonotone BFGS method was first studied by Liu, et al in [15]. Subsequently, two other nonmonotone BFGS methods were proposed for solving problem (1) in [12,16]. Note that convergence analysis in all these algorithms was proved under convex assumption on the objective function.…”