  Minimizing the sum of a convex function and a composite function appears in
various fields. The generalized Levenberg--Marquardt (LM) method, also known as
the prox-linear method, has been developed for such optimization problems. The
method iteratively solves strongly convex subproblems with a damping term. This
study proposes a new generalized LM method for solving the problem with a
smooth composite function. The method enjoys three theoretical guarantees:
iteration complexity bound, oracle complexity bound, and local convergence
under a H\"olderian growth condition. The local convergence results include
local quadratic convergence under the quadratic growth condition; this is the
first to extend the classical result for least-squares problems to a general
smooth composite function. In addition, this is the first LM method with both
an oracle complexity bound and local quadratic convergence under standard
assumptions. These results are achieved by carefully controlling the damping
parameter and solving the subproblems by the accelerated proximal gradient
method equipped with a particular termination condition. Experimental results
show that the proposed method performs well in practice for several instances,
including classification with a neural network and nonnegative matrix
factorization.
