Find the maximum of x 2 + y 1 − x 1 − y 2 if ( x 1 − 1 / 2 ) 2 + ( y 1 − 1 / 2 ) 2 = 1 / 4 and ( x 2 − 1 / 2 ) 2 + ( y 2 − 1 / 2 ) 2 = 1 / 4 and x 1 y 2 = y 1 x 2 and all variables are positive.Is there a way to do this with inequalities? What about Lagrange Multipliers?

Question

Find the maximum of      x    2    +      y    1    −      x    1    −      y    2  if   (      x    1    −  1      /    2      )    2    +  (      y    1    −  1      /    2      )    2    =  1      /    4 and   (      x    2    −  1      /    2      )    2    +  (      y    2    −  1      /    2      )    2    =  1      /    4 and       x    1        y    2    =      y    1        x    2   and all variables are positive.Is there a way to do this with inequalities? What about Lagrange Multipliers?

Johnathan Morse · Accepted Answer

Using Lagrange multipliers, that is to say minimizing  F  =      x    2    +      y    1    −      x    1    −      y    2    +  λ      (                  (                  x          1                −                  1          2                )            2        +                  (                  y          1                −                  1          2                )            2        −          1      4        )    +  μ      (                  (                  x          2                −                  1          2                )            2        +                  (                  y          2                −                  1          2                )            2        −          1      4        )    +  ν  (      x    1        y    2    −      y    1        x    2    )is a pure nightmare.What I did was to extract as much variables as I could do from the constraints that is to say      y          1      ±        =      1    2    ±            1      4        −                  (                  x          1                −                  1          2                )            2            y          2      ±        =      1    2    ±            1      4        −                  (                  x          2                −                  1          2                )            2            x    2    =            y      2              y      1            x    1  and considered all possible cases. For any combination of the   ±&#039;s,   F is now a function of       x    1   only and can be minimized (not pleasant but doable). Now, for each solution       x    1  , compute the value of   F.I shall not report all the results (too long to be typed) but, if I am not mistaken, the best point corresponds to      x    1    =      1    4          x    2    =            2      +              3              4          y    1    =            2      −              3              4          y    2    =      1    4        ⟹          F          m      a      x        =      1    2  I let you the pleasure of doing it.

Find the maximum of x 2 </msub> + y 1 </msub> −

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