Stochastic approximation type methods for constrained systems: Algorithms and numerical results

A stochastic version of the standard nonlinear programming problem is considered. A function
f(x)
is observed in the presence of noise, and we seek to minimize
f(x)
for
x \in C = {x:q^{i}(x) \leq 0}
, where
q^{i}(x)
are constraints. Numerous practical examples exist. Algorithms are discussed for selecting a sequence X
n
which converges wp 1 to a point where a necessary condition for optimality holds. The algorithms use, of course, noise-corrupted observations on the
f(x)
. Numerical results are presented. They indicate that the approach is quite versatile, and can be a useful tool for systematic Monte-Carlo optimization of constrained systems, a much-neglected area. However, many practical problems remain to be resolved, e.g., investigation of efficient one-dimensional search methods and of the tradeoffs between the effort spent per search cycle and the number of search cycles.