A question about the proof

[Jarvis-K]

I am reading about the WQMIX, but I am not sure about the proof in the appendix:

Why the $Q_{tot}^\prime \in Q^{mix}$ ? How can I check a construction that in the QMIX family or not?
Would u like to help me figure it out?

[SEUCGX]

Yes, I have the same problem. This conclusion seems to be unacceptable. Have you solved this problem？

[Jarvis-K]

The QMIX family can be easily checked as $\mathcal{Q}^{m i x}:={Q_{t o t} \mid Q_{t o t}(s, \mathbf{u})=f_s(Q_1(s, u_1), \ldots Q_n(s, u_n)), \frac{\partial f_s}{\partial Q_a} \geq 0, Q_a(s, u) \in \mathbb{R}}$.
The

ensures the $Q^{mix}$ family. It should be noted that for $\hat{u}$, $Q_{tot}(\hat{u}) \geq Q_{tot}(u)$ for any $u$ and the positive gradient requirements of qmix family will not be violated by increase the $Q_{tot}(\hat{u})$ to a larger value $Q(\hat{u})$ .