Let f ( x , y ) be a non-separable, non-negative real-valued function, that is jointly concave in x and y. We want to maximize f ( x , y ) over x and y.Is the sequential maximization max x max y f ( x , y )always equal to the simultaneous maximization max x , y f ( x , y )Or is there a simple counter-example for this?I know that if is separable, then the sequential maximization and simultaneous maximization are equal. Are there any conditions on a non-separable function such that this result still holds?

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Let   f  (  x  ,  y  ) be a non-separable, non-negative real-valued function, that is jointly concave in   x and   y. We want to maximize   f  (  x  ,  y  ) over   x and   y.Is the sequential maximization      max          x            max          y        f  (  x  ,  y  )always equal to the simultaneous maximization      max          x      ,             y        f  (  x  ,  y  )Or is there a simple counter-example for this?I know that if  is separable, then the sequential maximization and simultaneous maximization are equal. Are there any conditions on a non-separable function such that this result still holds?

Marlee Norman · Accepted Answer

Let   (      x    0    ,      y    0    ) be any two elements. Then clearly we have that   f  (      x    0    ,      y    0    )  ≤      max    y    f  (      x    0    ,  y  )Moreover we notice that       max    x        max    y    f  (  x  ,  y  ) is the maximum over elements of the form       max    y    f  (      x    ′    ,  y  ) for some       x    ′  . In particular this implies that   f  (      x    0    ,      y    0    )  ≤      max    y    f  (      x    0    ,  y  )  ≤      max    x        max    y    f  (  x  ,  y  ). As       x    0    ,      y    0   were chosen arbitrarily we may conclude that       max          x      ,      y        f  (  x  ,  y  )  ≤      max    x        max    y    f  (  x  ,  y  ) and the inequality in the other direction is obvious.

Leah Pope · Accepted Answer

Thanks! That&#039;s very helpful

Let f ( x , y ) be a non-separable, non-negative real-valued function, that is

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