Sequential Joint Detection and Estimation: Optimal Average Stopping Time

This paper considers a binary sequential joint detection and estimation problem, which aims to find an optimal scheme to minimize the average stopping time under the constraints on the detection error probabilities and the estimation errors. The scheme consists of a randomized stopping rule, a randomized detector and two estimators under two different decisions. To obtain the optimal scheme, we need to solve a non-convex minimization problem defined on a complicated Banach space. First, this non-convex problem is equivalently transformed into a concave maximization problem defined on a finite-dimensional Hilbert space, which however involves an objective function without closed-form expression. Then, by exploiting Danskin's theorem, the subdifferential of the objective function can be computed. Next, we propose a globally convergent algorithm based on the projected subgradient method to obtain the globally optimal solution of the convex problem. Last, by utilizing the solution of the convex problem, we provide a method to efficiently construct a solution of the original non-convex problem, i.e., the optimal randomized stopping rule, randomized detector and estimators. The numerical simulation results show the verifications of our methods.