Conditional Monte Carlo or pre-integration is a useful tool for reducing
variance and improving regularity of integrands when applying Monte Carlo and
quasi-Monte Carlo (QMC) methods. To choose the variable to pre-integrate with,
one need to consider both the variable importance and the tractability of the
conditional expectation. For integrals over a Gaussian distribution, one can
pre-integrate over any linear combination of variables. Liu and Owen (2022)
propose to choose the linear combination based on an active subspace
decomposition of the integrand. However, pre-integrating over such selected
direction might be intractable. In this work, we address this issue by finding
the active subspaces subject to the constraints such that pre-integration can
be easily carried out. The proposed method is applied to some examples in
derivative pricing under stochastic volatility models and is shown to
outperform previous methods