Probability Gradient Estimation by Set-Valued Calculus and Applications in Network Design

Let θ 7→ P (θ) be a set-valued mapping from R into the family of closed compact polyhedra in R. Let ξ be a R valued random variable. Many stochastic optimization problems in computer networking, system reliability, transportation, telecommunication, finance etc. can be formulated as a problem to minimize (or maximize) the probability P{ξ ∈ P (θ)} under some constraints on the decision variable θ. For a practical solution of such a problem, one has to approximate the objective function and its derivative by Monte Carlo simulation, since a closed analytical expression is only available in rare cases. In this paper, we present a new method of approximating the gradient of P{ξ ∈ P (θ)} w.r.t θ by sampling, which is based on the concept of set-wise derivative. Quite surprisingly, it turns out that it is typically easier to approximate the derivative than the objective itself.