  We present a new algorithm for computing pure-strategy
$\varepsilon$-Bayes-Nash equilibria ($\varepsilon$-BNEs) in combinatorial
auctions with continuous value and action spaces. An essential innovation of
our algorithm is to separate the algorithm's search phase (for finding the
$\varepsilon$-BNE) from the verification phase (for computing the
$\varepsilon$). Using this approach, we obtain an algorithm that is both very
fast and provides theoretical guarantees on the $\varepsilon$ it finds. Our
main technical contribution is a verification method which allows us to upper
bound the $\varepsilon$ across the whole continuous value space without making
assumptions about the mechanism. Using our algorithm, we can now compute
$\varepsilon$-BNEs in multi-minded domains that are significantly more complex
than what was previously possible to solve. We release our code under an
open-source license to enable researchers to perform algorithmic analyses of
auctions, to enable bidders to analyze different strategies, and to facilitate
many other applications.
