  Conditional Monte Carlo or pre-integration is a powerful tool for reducing
variance and improving the regularity of integrands when using Monte Carlo and
quasi-Monte Carlo (QMC) methods. To select the variable to pre-integrate, one
must consider both the variable's importance and the tractability of the
conditional expectation. For integrals over a Gaussian distribution, any linear
combination of variables can potentially be pre-integrated. Liu and Owen (2022)
propose to select the linear combination based on an active subspace
decomposition of the integrand. However, pre-integrating the selected direction
might be intractable. In this work, we address this issue by finding the active
subspace subject to constraints such that pre-integration can be easily carried
out. The proposed algorithm also provides a computationally-efficient
alternative to dimension reduction for pre-integrated functions. The method is
applied to examples from computational finance, density estimation, and
computational chemistry, and is shown to achieve smaller errors than previous
methods.
