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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 costis 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 valuesof the accepted demands, minus the convex cost?  We consider this problem in the random-order a.k.a. secretary model, and showan $O(d)$-competitive algorithm for the case where the convex cost function isalso supermodular. If the set of accepted demands must also be independent in agiven matroid, we give an $O(d^3 \alpha)$-competitive algorithm for thesupermodular case, and an improved $O(d^2\alpha)$ if the convex cost functionis also separable. Here $\alpha$ is the competitive ratio of the best algorithmfor the submodular secretary problem. These extend and improve previous resultsknown for this problem. Our techniques are simple but use powerful ideas fromconvex duality, which give clean interpretations of existing work, and allow usto give the extensions and improvements.

Maximizing Profit with Convex Costs in the Random-order Model