  This paper addresses the study of a new class of nonsmooth optimization
problems, where the objective is represented as a difference of two generally
nonconvex functions. We propose and develop a novel Newton-type algorithm to
solving such problems, which is based on the coderivative generated
second-order subdifferential (generalized Hessian) and employs advanced tools
of variational analysis. Well-posedness properties of the proposed algorithm
are derived under fairly general requirements, while constructive convergence
rates are established by using additional assumptions including the
Kurdyka--{\L}ojasiewicz condition. We provide applications of the main
algorithm to solving a general class of nonsmooth nonconvex problems of
structured optimization that encompasses, in particular, optimization problems
with explicit constraints. Finally, applications and numerical experiments are
given for solving practical problems that arise in biochemical models,
constrained quadratic programming, etc., where advantages of our algorithms are
demonstrated in comparison with some known techniques and results.
