Optimal Bidding, Allocation, and Budget Spending for a Demand-Side Platform with Generic Auctions

We develop an optimization model and corresponding algorithm for the management of a demand-side platform (DSP), whereby the DSP acquires valuable impressions for its advertiser clients in a real-time bidding environment. We propose a highly flexible model for the DSP to maximize its profit while maintaining acceptable levels of budget spending for its advertisers. Our model achieves flexibility and improved performance primarily through two different aspects: (i) we replace standard budget constraints with a more general utility function over budget spending levels, and (ii) we can accommodate arbitrary auction types by directly modeling the interactions between the DSP and the 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 obtain a convex dual problem that can be efficiently solved with subgradient based algorithms and from which a primal solution may be recovered efficiently. Under a natural and intuitive “increasing marginal cost” condition, as well as under a more general condition, we show that there is zero duality gap between the dual problem and the original non-convex primal problem. Under the same conditions, we also demonstrate convergence of our algorithm to an optimal solution of the non-convex formulation as the dual problem is solved to near optimality. We conduct experiments on both synthetic data as well as data from a real DSP, and our results demonstrate how our algorithm allows the DSP to better trade off between profitability and budget spending as compared to a widely used “greedy” heuristic approach