This paper proposes and develops a new Newton-type algorithm to solve
subdifferential inclusions defined by subgradients of extended-real-valued
prox-regular functions. The proposed algorithm is formulated in terms of the
second-order subdifferential of such functions that enjoys extensive calculus
rules and can be efficiently computed for broad classes of extended-real-valued
functions. Based on this and on metric regularity and subregularity properties
of subgradient mappings, we establish verifiable conditions ensuring
well-posedness of the proposed algorithm and its local superlinear convergence.
The obtained results are also new for the class of equations defined by
continuously differentiable functions with Lipschitzian derivatives
(C1,1 functions), which is the underlying case of our
consideration. The developed algorithm for prox-regular functions is formulated
in terms of proximal mappings related to and reduces to Moreau envelopes.
Besides numerous illustrative examples and comparison with known algorithms for
C1,1 functions and generalized equations, the paper presents
applications of the proposed algorithm to the practically important class of
Lasso problems arising in statistics and machine learning.Comment: 35 page