  In this paper we consider a class of structured monotone inclusion (MI)
problems that consist of finding a zero in the sum of two monotone operators,
in which one is maximal monotone while another is locally Lipschitz continuous.
In particular, we first propose a primal-dual extrapolation (PDE) method for
solving a structured strongly MI problem by modifying the classical
forward-backward splitting method by using a point and operator extrapolation
technique, in which the parameters are adaptively updated by a backtracking
line search scheme. The proposed PDE method is almost parameter-free, equipped
with a verifiable termination criterion, and enjoys an operation complexity of
${\cal O}(\log \epsilon^{-1})$, measured by the amount of fundamental
operations consisting only of evaluations of one operator and resolvent of
another operator, for finding an $\epsilon$-residual solution of the structured
strongly MI problem. We then propose another PDE method for solving a
structured non-strongly MI problem by applying the above PDE method to
approximately solve a sequence of structured strongly MI problems. The
resulting PDE method is parameter-free, equipped with a verifiable termination
criterion, and enjoys an operation complexity of ${\cal O}(\epsilon^{-1}\log
\epsilon^{-1})$ for finding an $\epsilon$-residual solution of the structured
non-strongly MI problem. As a consequence, we apply the latter PDE method to
convex conic optimization, conic constrained saddle point, and variational
inequality problems, and obtain complexity results for finding an
$\epsilon$-KKT or $\epsilon$-residual solution of them under local Lipschitz
continuity. To the best of our knowledge, no prior studies were conducted to
investigate methods with complexity guarantees for solving the aforementioned
problems under local Lipschitz continuity. All the complexity results obtained
in this paper are entirely new.
