jmc

We may think of trying to apply AM-GM directly to all five terms. Ignoring the constants, this give us a term of $$\sqrt[5]{x^2\cdot xy\cdot y^2\cdot yz\cdot z^2}=\sqrt[5]{x^3{y}^4{z}^3}.$$This doesn't work, because the condition is $xyz=\frac{2}{3},$ so we want a power of $xyz.$ So, to get more one power of $y,$ relative to $x$ and $z,$ we split every term except $y^2$ in half: $$\frac{x^2}{2}+\frac{x^2}{2}+3xy+3xy+18y^2+6yz+6yz+2z^2+2z^2.$$Then by AM-GM, $$\begin{array}{rl}& \frac{x^2}{2}+\frac{x^2}{2}+3xy+3xy+18y^2+6yz+6yz+2z^2+2z^2\\ & \ge 9\sqrt[9]{\frac{x^2}{2}\cdot \frac{x^2}{2}\cdot 3xy\cdot 3xy\cdot 18y^2\cdot 6yz\cdot 6yz\cdot 2z^2\cdot 2z^2}\\ & =9\sqrt[9]{5832x^6{y}^6{z}^6}\\ & =18.\end{array}$$Equality occurs when $\frac{x^2}{2}=3xy=18y^2=6yz=2z^2.$ Along with the condition $xyz=\frac{2}{3},$ we can solve to get $x=2,$ $y=\frac{1}{3},$ $z=1,$ so the minimum value is $\boxed{18}.$