1. A computer-implemented method comprising the steps of: automatically projecting user behavior pertaining to a display of a plurality of items during a specified future time period; wherein the step of automatically projection includes generating a solution to an optimization problem that is based on a relaxed formula that is a version of an original formula in which an original constraint from the original formula is replaced by a relaxed constraint in the relaxed formula; wherein the relaxed constraint is less restrictive than the original constraint; and based on the solution to the optimization problem, selecting a particular item of the plurality of items for display during the specified future time period; wherein the method is performed by one or more computing devices.