  Existing decentralized coded caching solutions cannot guarantee small loads
in the general scenario with arbitrary file sizes and cache sizes. In this
paper, we propose an optimization framework for decentralized coded caching in
the general scenario to minimize the worst-case load and average load (under an
arbitrary file popularity), respectively. Specifically, we first propose a
class of decentralized coded caching schemes for the general scenario, which
are specified by a general caching parameter and include several known schemes
as special cases. Then, we optimize the caching parameter to minimize the
worst-case load and average load, respectively. Each of the two optimization
problems is a challenging nonconvex problem with a nondifferentiable objective
function. For each optimization problem, we develop an iterative algorithm to
obtain a stationary point using techniques for solving Complementary Geometric
Programming (GP). We also obtain a low-complexity approximate solution by
solving an approximate problem with a differentiable objective function which
is an upper bound on the original nondifferentiable one, and characterize the
performance loss caused by the approximation. Finally, we present two
information-theoretic converse bounds on the worst-case load and average load
(under an arbitrary file popularity) in the general scenario, respectively. To
the best of our knowledge, this is the first work that provides
optimization-based decentralized coded caching schemes and
information-theoretic converse bounds for the general scenario.
