Solving deterministic problems by stochastic approximation

AbstractStochastic approximation originally proposed by Robbins and Monro for stochastic problems is shown to be an effective computational tool for the numerical solution of deterministic problems. Some of the striking features of this approach are: Convergence can be proved for a general function and no differentiations or partial derivatives are needed. With some modifications of the required constant for deterministic problems, convergence can be accelerated considerably. To illustrate the approach, both algebraic equations and 2-point boundary value problems in ordinary differential equations are solved. Although these two classes of problems are quite different, they can be solved by essentially the same iterative approach. This shows the advantage and the generality of the method. Only 1-dimensional problems are investigated. Multidimensional problems and optimization problems with and without constraints will be discussed in other papers