  In this paper, we examine the Sample Average Approximation (SAA) procedure
within a framework where the Monte Carlo estimator of the expectation is
biased. We also introduce Multilevel Monte Carlo (MLMC) in the SAA setup to
enhance the computational efficiency of solving optimization problems. In this
context, we conduct a thorough analysis, exploiting Cram\'er's large deviation
theory, to establish uniform convergence, quantify the convergence rate, and
determine the sample complexity for both standard Monte Carlo and MLMC
paradigms. Additionally, we perform a root-mean-squared error analysis
utilizing tools from empirical process theory to derive sample complexity
without relying on the finite moment condition typically required for uniform
convergence results. Finally, we validate our findings and demonstrate the
advantages of the MLMC estimator through numerical examples, estimating
Conditional Value-at-Risk (CVaR) in the Geometric Brownian Motion and nested
expectation framework.
