  In many practical applications, heuristic or approximation algorithms are
used to efficiently solve the task at hand. However their solutions frequently
do not satisfy natural monotonicity properties of optimal solutions. In this
work we develop algorithms that are able to restore monotonicity in the
parameters of interest. Specifically, given oracle access to a (possibly
non-monotone) multi-dimensional real-valued function $f$, we provide an
algorithm that restores monotonicity while degrading the expected value of the
function by at most $\varepsilon$. The number of queries required is at most
logarithmic in $1/\varepsilon$ and exponential in the number of parameters. We
also give a lower bound showing that this exponential dependence is necessary.
Finally, we obtain improved query complexity bounds for restoring the weaker
property of $k$-marginal monotonicity. Under this property, every
$k$-dimensional projection of the function $f$ is required to be monotone. The
query complexity we obtain only scales exponentially with $k$.
