How do I maximize this function subject to this constraint?

$\max (\min(2x_A,y_A)+\min(x_B,2y_B))$ subject to $x_A+x_B=1$ and $y_A+y_B=1$

  • $\begingroup$ 4/3 is the answer? $\endgroup$ – Win Vineeth Sep 10 '16 at 3:47

By symmetry, $2x_A=y_A$ and $x_B=2y_B$ Solving, you get $x_A=\frac 13$ and $y_A=\frac 23$

Thus, answer is $\frac 43$

  • $\begingroup$ could you explain why we use symmetry in this question? $\endgroup$ – Rainroad Sep 10 '16 at 16:23
  • $\begingroup$ The minimum or maximum occurs only in two situations in linear problems - The symmetrical and the Extreme. You can try out the extreme ($x_A=1$ and $y_B=1$) It doesn't give maximum. $\endgroup$ – Win Vineeth Sep 10 '16 at 23:58
  • $\begingroup$ could you explain why this is a linear problem? $\endgroup$ – Rainroad Sep 11 '16 at 3:49
  • $\begingroup$ $x_A$ and $x_B$ are linearly related. $y_A$ and $y_B$ are linearly related. Min and Max are linear functions as long as their components are linear. Therefore, the total function is a linear function with respect to $x_A$ and $y_B$. $\endgroup$ – Win Vineeth Sep 11 '16 at 11:24

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