An extension of the Gauss-Newton algorithm is proposed to find local minimizers of penalized nonlinear least squares problems, under generalized Lipschitz assumptions. Convergence results of local type are obtained, as well as an estimate of the radius of the convergence ball. Some applications for solving constrained nonlinear equations are discussed and the numerical performance of the method is assessed on some significant test problems.
@cite_14: An extension of the Gauss—Newton method for nonlinear equations to convex composite optimization is described and analyzed. Local quadratic convergence is established for the minimization ofh ο F under two conditions, namelyh has a set of weak sharp minima,C, and there is a regular point of the inclusionF(x) ∈ C. This result extends a similar convergence result due to Womersley (this journal, 1985) which employs the assumption of a strongly unique solution of the composite functionh ο F. A backtracking line-search is proposed as a globalization strategy. For this algorithm, a global convergence result is established, with a quadratic rate under the regularity assumption.
@cite_26: The local quadratic convergence of the Gauss-Newton method for convex composite optimization f=h∘F is established for any convex function h with the minima set C, extending Burke and Ferris’ results in the case when C is a set of weak sharp minima for h.
@cite_8: We introduce a notion of quasi regularity for points with respect to the inclusion @math , where @math is a nonlinear Frechet differentiable function from @math to @math . When @math is the set of minimum points of a convex real-valued function @math on @math and @math satisfies the @math -average Lipschitz condition of Wang, we use the majorizing function technique to establish the semilocal linear quadratic convergence of sequences generated by the Gauss-Newton method (with quasi-regular initial points) for the convex composite function @math . Results are new even when the initial point is regular and @math is Lipschitz.
@cite_48: Nonsmooth optimization problems are divided into two categories. The first is composite nonsmooth problems where the generalized gradient can be approximated by information available at the current point. The second is basic nonsmooth problems where the generalized gradient must be approximated using information calculated at previous iterates.
@cite_32: This paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionF(x)=f(x)+h(c(x)) wheref andc are smooth buth(c) may be nonsmooth. Local convergence at a second order rate is established for the generalized Gauss—Newton method whenh is convex and globally Lipschitz and the minimizer is strongly unique. Local convergence at a second order rate is established for a generalized Newton method when the minimizer satisfies nondegeneracy, strict complementarity and second order sufficiency conditions. Assuming the minimizer satisfies these conditions, necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established. Necessary and sufficient conditions for a two-step superlinear rate of convergence are also established when only reduced curvature information is available. All these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems.
@cite_10: 
Problem @math can be cast, in principle, as a composite optimization problem of the form eq:conv_comp x X , h(c(x)), by setting @math and @math as follows: eq:h_and_c h(s,v)= s ^2+J(v), c(x)=(F(x)-y,x). Problem has been deeply studied in @cite_14 @cite_10 @cite_8 @cite_26 @cite_32 @cite_48 from different point of views, mainly in the case @math and @math , but the hypotheses significantly differ, as well as the obtained results. More specifically, in @cite_10 , the assumptions are too general to capture the features of problem @math , and allow only to get convergence results much weaker than the ones obtained in Theorem . Regarding all the remaining papers, as a matter of fact, the following special case of inclusion problem is treated eq:conv_incl c(x) C, C = h. In particular, the existence of an @math such that @math is always assumed. That hypothesis is of course reasonable if we think of @math as a kind of norm, but if we take it as in , then @math and we are lead to the condition [ ,x X, F(x)=y and x J ] which is too demanding for our original problem @math .