  Searches for non-resonant and resonant Higgs boson pair production are
performed in the $\gamma\gamma WW^{*}$ channel with the final state of
$\gamma\gamma\ell\nu jj$ using 36.1 fb$^{-1}$ of proton-proton collision data
recorded at a centre-of-mass energy of $\sqrt{s} = 13$ TeV by the ATLAS
detector at the Large Hadron Collider. No significant deviation from the
Standard Model prediction is observed. A 95% confidence-level observed upper
limit of 7.7 pb is set on the cross section for non-resonant production, while
the expected limit is 5.4 pb. A search for a narrow-width resonance $X$
decaying to a pair of Standard Model Higgs bosons $HH$ is performed with the
same set of data, and the observed upper limits on $\sigma(pp \rightarrow X)
\times B(X \rightarrow HH)$ range between 40.0 pb and 6.1 pb for masses of the
resonance between 260 GeV and 500 GeV, while the expected limits range between
17.6 pb and 4.4 pb. When deriving the limits above, the Standard Model
branching ratios of the $H \rightarrow \gamma\gamma$ and $H \rightarrow WW^{*}$
are assumed.
