  This paper considers distributed online optimization with time-varying
coupled inequality constraints. The global objective function is composed of
local convex cost and regularization functions and the coupled constraint
function is the sum of local convex functions. A distributed online primal-dual
dynamic mirror descent algorithm is proposed to solve this problem, where the
local cost, regularization, and constraint functions are held privately and
revealed only after each time slot. Without assuming Slater's condition, we
first derive regret and constraint violation bounds for the algorithm and show
how they depend on the stepsize sequences, the accumulated dynamic variation of
the comparator sequence, the number of agents, and the network connectivity. As
a result, under some natural decreasing stepsize sequences, we prove that the
algorithm achieves sublinear dynamic regret and constraint violation if the
accumulated dynamic variation of the optimal sequence also grows sublinearly.
We also prove that the algorithm achieves sublinear static regret and
constraint violation under mild conditions. Assuming Slater's condition, we
show that the algorithm achieves smaller bounds on the constraint violation. In
addition, smaller bounds on the static regret are achieved when the objective
function is strongly convex. Finally, numerical simulations are provided to
illustrate the effectiveness of the theoretical results.
