Adaptive optimization of the Monte-Carlo method (Q1842446)

Let \(\zeta(u)\) be a random nonnegative function where \(u\in U\) is from a closed convex bounded subset in a finite-dimensional Euclidean space. Assume that the mean function \(m(u)= E(\zeta(u))\) is a continuously differentiable convex function. It is constructed a sequence of random vectors \(u^j\) with \(P\{\lim_{k\to \infty} {1\over k} \sum^k_{j= 1} (\zeta(u^j))= m^*\}= 1\), where \(m^*= \min\{m(u): u\in U\}\) and a bound rate of convergence in probability is proved.