Branch and bound stanford slides doubt

On the 6th slide at https://web.stanford.edu/class/ee364b/lectures/bb_slides.pdf, while defining L2 and U2, why are we taking min for both?

• Please be more explicit with your question: in particular, giving the general context and the definitions of $L_2$ and $U_2$ only takes a few lines, and makes this question potentially useful to more people than just yourself. – cody Oct 7 '15 at 15:07
• Please edit the question to make the question self-contained and provide all relevant context (the definitions of L2 and U2, etc.). People shouldn't have to click to look on a separate page to understand what you are asking; and if that link stops working, we want the question to still be understandable. Finally, what are your thoughts? – D.W. Oct 8 '15 at 17:42

The slides are correct: remember that we are trying to find a bound on the optimal value of $f$, which is defined as $$\Phi_\mbox{min}({\cal Q}_\mbox{init}) =\min_{x\in{\cal Q}_{\tiny\mbox{init}}}f(x)$$
To find that minimum, we approximate it by upper and lower bounds $\Phi_\mbox{ub}$ and $\Phi_{\mbox{lb}}$. Let's concentrate on $\Phi_{\mbox{ub}}$. Note that if ${\cal Q}={\cal Q}_1\cup{\cal Q}_2$, then both $$\min(\Phi_{\mbox{ub}}(\cal Q_1),\Phi_{\mbox{ub}}(\cal Q_2))\geq\Phi_\mbox{min}(\cal Q)$$ and $$\max(\Phi_{\mbox{ub}}(\cal Q_1),\Phi_{\mbox{ub}}(\cal Q_2))\geq\Phi_\mbox{min}(\cal Q)$$
But we might as well take $\min$, since that is the value that is closer to the global minimum $\Phi_\min(\cal Q)$!
In fact, taking the minimum is going to be necessary if we want the branch and bound method to converge to the minimum as we sub-divide $\cal Q$ into smaller and smaller pieces.