  Suppose a set of requests arrives online: each request gives some value $v_i$
if accepted, but requires using some amount of each of $d$ resources. Our cost
is a convex function of the vector of total utilization of these $d$ resources.
Which requests should be accept to maximize our profit, i.e., the sum of values
of the accepted demands, minus the convex cost?
  We consider this problem in the random-order a.k.a. secretary model, and show
an $O(d)$-competitive algorithm for the case where the convex cost function is
also supermodular. If the set of accepted demands must also be independent in a
given matroid, we give an $O(d^3 \alpha)$-competitive algorithm for the
supermodular case, and an improved $O(d^2\alpha)$ if the convex cost function
is also separable. Here $\alpha$ is the competitive ratio of the best algorithm
for the submodular secretary problem. These extend and improve previous results
known for this problem. Our techniques are simple but use powerful ideas from
convex duality, which give clean interpretations of existing work, and allow us
to give the extensions and improvements.
