  This paper considers the efficient minimization of the infinite time average
of a stationary ergodic process in the space of a handful of design parameters
which affect it. Problems of this class, derived from physical or numerical
experiments which are sometimes expensive to perform, are ubiquitous in
engineering applications. In such problems, any given function evaluation,
determined with finite sampling, is associated with a quantifiable amount of
uncertainty, which may be reduced via additional sampling. The present paper
proposes a new optimization algorithm to adjust the amount of sampling
associated with each function evaluation, making function evaluations more
accurate (and, thus, more expensive), as required, as convergence is
approached. The work builds on our algorithm for Delaunay-based Derivative-free
Optimization via Global Surrogates ($\Delta$-DOGS). The new algorithm, dubbed
$\alpha$-DOGS, substantially reduces the overall cost of the optimization
process for problems of this important class. Further, under certain
well-defined conditions, rigorous proof of convergence to the global minimum of
the problem considered is established.
