Optimal dynamic advertising for revenue management with budget constraint

We consider a single-product revenue management problem where given an initial inventory vector, the objective is to maximize expected revenues by dynamically spending on advertisement over a finite sales horizon. At any epoch, there is an advertising budget constraint. We derive the Hamilton-Jacobi sufficient conditions for the profit-to-go function. Based on the optimality conditions, we show the structure properties of the optimal advertising strategy. Because computing the optimal policy is very challenging, a simple even-level heuristic policy is provided.