## Profit Maximization for Online Advertising Demand-Side Platforms

#### Paul Grigas, Alfonso Lobos, Zheng Wen, and Kuang-chih Lee

We develop an optimization model and corresponding algorithm for the management of a demand-side platform (DSP), whereby the DSP aims to maximize its own profit while acquiring valuable impressions for its advertiser clients. We formulate the problem of profit maximization for a DSP interacting with ad exchanges in a real-time bidding environment in a cost-per-click/cost-per-action pricing model. Our proposed formulation leads to a nonconvex optimization problem due to the joint optimization over both im- pression allocation and bid price decisions. We use Lagrangian relaxation to develop a tractable convex dual problem, which, due to the properties of second-price auctions, may be solved efficiently with subgradient methods. We propose a two-phase solution proce- dure, whereby in the first phase we solve the convex dual problem using a subgradient algorithm, and in the second phase we use the previously computed dual solution to set bid prices and then solve a linear optimization problem to obtain the allocation probability variables. On several synthetic examples, we demonstrate that our proposed solution approach leads to superior performance over a baseline method that is used in practice.