Local convergence of the Levenberg–Marquardt method under Hölder metric subregularity

Abstract

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under Hölder metric subregularity of the function defining the equation and Hölder continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the Lojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the Lojasiewicz gradient inequality assumption.Keywords Nonlinear equation · Levenberg-Marquardt method · Local convergence rate · Hölder metric subregularity · Lojasiewicz inequality Mathematics Subject Classification (2010) 65K05 · 65K10 · 90C26 · 92C42 F.J.

Finding zeros of Hölder metrically subregular mappings via globally convergent Levenberg–Marquardt methods

“…Such an inequality is referred as an error bound (Lipschitzian error bound or metric regularity) condition. The notion of error bound has been very popular during the last few decades to study the local convergence of optimisation methodologies; however, there are many important mappings where (3) is not satisfied , see, e.g., [4,34]. This motivated the authors of [4] to propose a weaker condition so-called the Hölder metric subregularity (Hölderian error bound), i.e.,…”

“…for δ ∈ ]0, 1] and r ∈ ]0, 1[. There are many mappings satisfying this condition, see, e.g., [4,34] and references therein. See also Section 5 for a real-world nonlinear system satisfying (4), but not (3).…”

“…The choice of the parameter µ k has substantial impacts on the global convergence, the local convergence rate, and the computational efficiency of Levenberg-Marquardt methods, cf. [4,31,38,47,48]. Hence, several ways to specify and to adapt this parameter have been proposed; see, e.g., [18,19,48].…”

“…Hence, several ways to specify and to adapt this parameter have been proposed; see, e.g., [18,19,48]. A recently proposed Levenberg-Marquardt method by the authors [4] suggests an adaptive parameter µ k based on the order δ ∈ ]0, 1] of the Hölder metric subregularity (4), i.e.,…”

“…where η ∈ ]0, 4δ[, ξ k ∈ [ξ min , ξ max ] and ω k ∈ [ω min , ω max ] with ξ min + ω min > 0. In [4], this Levenberg-Marquardt method, with adaptive regularisation (LM-AR), was presented and its local convergence was studied for Hölder metrically subregular mappings. If one assumes that the starting point x 0 is close enough to a solution x * of (2), then the Levenberg-Marquardt method is known to be quadratically convergent if ∇h(x * ) is nonsingular, in which case it is clearly convergent to a solution to (1).…”