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Under the hypothesis that an initial point is a quasi-regular point, we use a majorant condition to present a new semilocal 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.

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