Convergence of the Gauss--Newton Method for Convex Composite Optimization under a Majorant Condition

Abstract

Under the hypothesis that an initial point is a quasi-regular point, we use a majorant condition to present a new semi-local convergence analysis of an extension of the Gauss-Newton method for solving convex composite optimization problems. In this analysis the conditions and proof of convergence are simplified by using a simple majorant condition to define regions where a Gauss-Newton sequence is "well behaved". AMSC: 47J15, 65H10.

Expanding the applicability of the Gauss–Newton method for convex optimization under a majorant condition

“…3. Numerical examples and applications of our theoretical results and favorable comparisons to earlier studies (see, e.g., [12,17,18,21,22]) are presented in Sect. 4.…”

“…Recently, in the elegant studies on the Gauss-Newton method (GNM) by Li, Ng (see, e.g., [21,22]) and Ferreira et al [17], the notion of quasi-regularity for x 0 ∈ R n with respect to inclusion problem was used. This notion generalizes the case of regularity studied in the seminal paper by Burke and Ferris (see, e.g., [12]).…”

“…This notion generalizes the case of regularity studied in the seminal paper by Burke and Ferris (see, e.g., [12]). The regularity condition was inaugurated by Robinson in [24] (see also, e.g., [14][15][16][17][18][19]). …”

“…In this paper, motivated by the work in [17] and optimization considerations, we present a convergence analysis of Gauss-Newton method (defined by Algorithm (GNA) in Sect. 2).…”

“…2 contains the definition of (GNA). In order for us to make the paper as self contained as possible, the notion of quasi-regularity is also re-introduced (see, e.g., [12,17,21]). The semilocal convergence analysis of (GNA) is presented in Sect.…”