I have encountered that in some cases maximization of a function had been substituted with a maximization of its monotone transformation.For example, finding the min or max of f ( x , y ) = ( ( x − 1 ) 2 + ( y − 2 ) 2 ) 1 / 2 is the same as finding the min / max of ( x 2 + y 2 ) and then take the square root of the answer. So, my questions are:1) Is it true for any monotone transformation g?2) Is the same principle allow us to maximize /minimize log ⁡ g ( x ) instead of g ( x ) itself?

Question

I have encountered that in some cases maximization of a function had been substituted with a maximization of its monotone transformation.For example, finding the min or max of   f  (  x  ,  y  )  =  (  (  x  −  1      )    2    +  (  y  −  2      )    2        )          1              /            2       is the same as finding the min / max of   (      x    2    +      y    2    ) and then take the square root of the answer. So, my questions are:1) Is it true for any monotone transformation   g?2) Is the same principle allow us to maximize /minimize   log  ⁡  g  (  x  ) instead of   g  (  x  ) itself?

Cynthia Herrera · Accepted Answer

It is true and, if the optimal is not on the boundary, it doesn&#039;t need to be monotonic.f objective function is   f  (  x  ) and a transformation is made to the variable/s   x  =  g  (  y  ).            d      f      (      g      (      y      )      )              d      y        =      f                          ′              (  x  )      g                          ′              (  y  )                    d        2            f      (      g      (      y      )      )              d              y        2              =      f                          ″              (  x  )  (      g                          ′                  )    2    +      f                          ′                  g                          ″              (  y  )at the optimal point       x    0        f                          ′              (      x    0    )  =  0therefore,                    d        2            f      (      g      (      y      )      )              d              y        2              =      f                          ″              (  x  )  (      g                          ′                  )    2  So if it the minimum/maximum of f, it should be min/max of   f  (  g  (  y  )  ). At this time, both functions are assumed to be derivative.

I have encountered that in some cases maximization of a function had been substituted with a maximiz

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