Carly Cannon

2022-07-02

I have the following equations and inequalities:
$1={A}^{\prime }+{B}^{\prime }$
$1=A+B+C$
$A\le {A}^{\prime }$
$B\le {B}^{\prime }$
All variables are bounded below by zero and above by one.
I wonder if I can find an analytic expression for the upper and lower bounds for the difference ${A}^{\prime }\left(1-C\right)-A$

lofoptiformfp

Expert

Step 1
First note ${A}^{\prime }\left(1-C\right)-A={A}^{\prime }\left(A+B\right)-A$
For the upper bound clearly we need to take $A=0$ , $B={B}^{\prime }$ and $C={A}^{\prime }$ . In which case the problem comes down to finding the maximum value of ${A}^{\prime }{B}^{\prime }={A}^{\prime }-{A}^{\prime 2}$ which is obtained if ${A}^{\prime }=\frac{1}{2}$ so an upper bound is $\frac{1}{4}$. For a lower bound clearly we need to take $A={A}^{\prime }$ , $B=0$ and $C={B}^{\prime }$ , so we have to find the minimum value of ${A}^{2}-A$ for $0\le A\le 1$ , which is obtained if $A=\frac{1}{2}$. So a lower bound is $-\frac{1}{4}$ .
So $-\frac{1}{4}\le {A}^{\prime }\left(1-C\right)-A\le \frac{1}{4}$ .

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