  We investigate a novel characteristic of the conjugate function associated to
a generic convex optimization problem which can subsequently be leveraged for
efficient dual decomposition methods. In particular, under mild assumptions, we
show a specific region of the domain of the conjugate function where there is
always a ray originating from any point of the region such that the gradients
of the conjugate remain constant along the ray. We refer to this characteristic
as a fixed gradient over rays (FGOR). We further show that the dual function
inherits from the conjugate the characteristic of FGOR. Then we provide a
thorough exposition of applying the FGOR characteristic to dual subgradient
methods. More importantly, we leverage FGOR to devise a simple stepsize rule
that can be prepended with state-of-the-art stepsize methods enabling them to
be more efficient. Furthermore, we investigate how the characteristic of FGOR
is used when solving the global consensus problem. We show that FGOR can be
exploited not only to expedite the convergence of the dual decomposition
methods but also to reduce the communication overhead when distributed
implementations are sought. Numerical experiments using quadratic objectives
and a regularized linear regression with a real data set are conducted to
compare the practical performance of FGOR with state-of-the-art stepsize
methods. Results show that the proposed approach can significantly improve the
convergence of existing methods while saving a considerable amount of
communication overhead.
