Making Decisions via Simulation

Here we will consider the problem min f ( ) θ∈Θ θ where, for a given value of θ , we are not able to evaluate analytically or numerically, but must obtain (noisy) estimates of f ( f ( ) θ ) θ using simulation. We will assume for now that the set of possible solutions Θ is uncountably infinite. In particular, suppose that Θ is an interval [ θ θ, θ Θ ] of real numbers. One approach to solving the problem is to estimate ) ( f min ∈ θ ) ( ' f θ using simulation and then use an iterative method to solve the equation f ) ( ' θ = 0. The most basic such method is called the Robbins-Monro Algorithm, and can be viewed as a stochastic version of Newton’s method for solving nonlinear equations.