  We develop a novel optimization model to maximize the profit of a Demand-Side
Platform (DSP) while ensuring that the budget utilization preferences of the
DSP's advertiser clients are adequately met. Our model is highly flexible and
can be applied in a Real-Time Bidding environment (RTB) with arbitrary auction
types, e.g., both first and second price auctions. Our proposed formulation
leads to a non-convex optimization problem due to the joint optimization over
both impression allocation and bid price decisions. Using Fenchel duality
theory, we construct a dual problem that is convex and can be solved
efficiently to obtain feasible bidding prices and allocation variables that can
be deployed in a RTB setting. With a few minimal additional assumptions on the
properties of the auctions, we demonstrate theoretically that our
computationally efficient procedure based on convex optimization principles is
guaranteed to deliver a globally optimal solution. We conduct experiments using
data from a real DSP to validate our theoretical findings and to demonstrate
that our method successfully trades off between DSP profitability and budget
utilization in a simulated online environment.
