I have the following equations and inequalities: 1 = A ′ + B ′ 1 = A + B + C A ≤ A ′ B ≤ B ′ 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 ′ ( 1 − C ) − A

Question

I have the following equations and inequalities:  1  =      A    ′    +      B    ′    1  =  A  +  B  +  C  A  ≤      A    ′    B  ≤      B    ′  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    ′    (  1  −  C  )  −  A

lofoptiformfp · Accepted Answer

Step 1First note       A    ′    (  1  −  C  )  −  A  =      A    ′    (  A  +  B  )  −  AFor the upper bound clearly we need to take   A  =  0 ,   B  =      B    ′   and   C  =      A    ′   . In which case the problem comes down to finding the maximum value of       A    ′        B    ′    =      A    ′    −      A          ′              2             which is obtained if       A    ′    =      1    2   so an upper bound is       1    4  . For a lower bound clearly we need to take   A  =      A    ′   ,   B  =  0 and   C  =      B    ′   , so we have to find the minimum value of       A          2        −  A for   0  ≤  A  ≤  1 , which is obtained if   A  =      1    2  . So a lower bound is   −      1    4   .So   −      1    4    ≤      A    ′    (  1  −  C  )  −  A  ≤      1    4   .

I have the following equations and inequalities: 1 = A ′ </msup>

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