  Myerson derived a simple and elegant solution to the single-parameter
revenue-maximization problem in his seminal work on optimal auction design
assuming the usual model of quasi-linear utilities. In this paper, we consider
a slight generalization of this usual model---from linear to convex "perceived"
payments. This more general problem does not appear to admit a solution as
simple and elegant as Myerson's.
  While some of Myerson's results extend to our setting, like his payment
formula (suitably adjusted), others do not. For example, we observe that the
solutions to the Bayesian and the robust (i.e., non-Bayesian) optimal auction
design problems in the convex perceived payment setting do not coincide like
they do in the case of linear payments. We therefore study the two problems in
turn.
  We derive an upper and a heuristic lower bound on expected revenue in our
setting. These bounds are easily computed pointwise, and yield monotonic
allocation rules, so can be supported by Myerson payments (suitably adjusted).
In this way, our bounds yield heuristics that approximate the optimal robust
auction, assuming convex perceived payments.
  We close with experiments, the final set of which massages the output of one
of the closed-form heuristics for the robust problem into an extremely fast,
near-optimal heuristic solution to the Bayesian optimal auction design problem.
