\[\begin{align}
& A={{x}^{2}}+2{{y}^{2}}+2xy-2\sqrt{2}x-2(\sqrt{2}+1)y+2022 \\
& ={{(x+y)}^{2}}-2\sqrt{2}(x+y)+({{y}^{2}}-2y+1)+2017 \\
& ={{(x+y-\sqrt{2})}^{2}}+{{(y-1)}^{2}}+2017\ge 2017 \\
& ''=''\Leftrightarrow y=1;x=\sqrt{2}-1 \\
& \\
\end{align}\]