  We study dual-based algorithms for distributed convex optimization problems
over networks, where the objective is to minimize a sum $\sum_{i=1}^{m}f_i(z)$
of functions over in a network. We provide complexity bounds for four different
cases, namely: each function $f_i$ is strongly convex and smooth, each function
is either strongly convex or smooth, and when it is convex but neither strongly
convex nor smooth. Our approach is based on the dual of an appropriately
formulated primal problem, which includes a graph that models the communication
restrictions. We propose distributed algorithms that achieve the same optimal
rates as their centralized counterparts (up to constant and logarithmic
factors), with an additional optimal cost related to the spectral properties of
the network. Initially, we focus on functions for which we can explicitly
minimize its Legendre-Fenchel conjugate, i.e., admissible or dual friendly
functions. Then, we study distributed optimization algorithms for non-dual
friendly functions, as well as a method to improve the dependency on the
parameters of the functions involved. Numerical analysis of the proposed
algorithms is also provided.
