Simulation-based optimization over discrete sets with noisy constraints

We consider a constrained optimization problem over a discrete set where noise-corrupted observations of the objective and constraints are available. The problem is challenging because the feasibility of a solution cannot be known for certain, due to the noisy measurements of the constraints. To tackle this issue, we propose a new method that converts constrained optimization into the unconstrained optimization problem of finding a saddle point of the Lagrangian. The method applies stochastic approximation to the Lagrangian in search of the saddle point. The proposed method is shown to converge, under suitable conditions, to the optimal solution almost surely (a.s.) as the number of iterations grows. We present the effectiveness of the proposed method numerically in two settings: (1) inventory control in a periodic review system, and (2) staffing in a call center.