  In this paper we develop accelerated first-order methods for convex
optimization with locally Lipschitz continuous gradient (LLCG), which is beyond
the well-studied class of convex optimization with Lipschitz continuous
gradient. In particular, we first consider unconstrained convex optimization
with LLCG and propose accelerated proximal gradient (APG) methods for solving
it. The proposed APG methods are equipped with a verifiable termination
criterion and enjoy an operation complexity of ${\cal O}(\varepsilon^{-1/2}\log
\varepsilon^{-1})$ and ${\cal O}(\log \varepsilon^{-1})$ for finding an
$\varepsilon$-residual solution of an unconstrained convex and strongly convex
optimization problem, respectively. We then consider constrained convex
optimization with LLCG and propose an first-order proximal augmented Lagrangian
method for solving it by applying one of our proposed APG methods to
approximately solve a sequence of proximal augmented Lagrangian subproblems.
The resulting method is equipped with a verifiable termination criterion and
enjoys an operation complexity of ${\cal O}(\varepsilon^{-1}\log
\varepsilon^{-1})$ and ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ for
finding an $\varepsilon$-KKT solution of a constrained convex and strongly
convex optimization problem, respectively. All the proposed methods in this
paper are parameter-free or almost parameter-free except that the knowledge on
convexity parameter is required. In addition, preliminary numerical results are
presented to demonstrate the performance of our proposed methods. To the best
of our knowledge, no prior studies were conducted to investigate accelerated
first-order methods with complexity guarantees for convex optimization with
LLCG. All the complexity results obtained in this paper are new.
