  Online optimization covers problems such as online resource allocation,
online bipartite matching, adwords (a central problem in e-commerce and
advertising), and adwords with separable concave returns. We analyze the worst
case competitive ratio of two primal-dual algorithms for a class of online
convex (conic) optimization problems that contains the previous examples as
special cases defined on the positive orthant. We derive a sufficient condition
on the objective function that guarantees a constant worst case competitive
ratio (greater than or equal to $\frac{1}{2}$) for monotone objective
functions. We provide new examples of online problems on the positive orthant
and the positive semidefinite cone that satisfy the sufficient condition. We
show how smoothing can improve the competitive ratio of these algorithms, and
in particular for separable functions, we show that the optimal smoothing can
be derived by solving a convex optimization problem. This result allows us to
directly optimize the competitive ratio bound over a class of smoothing
functions, and hence design effective smoothing customized for a given cost
function.
