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

micelije1mw

micelije1mw

Answered question

2022-05-03

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?

Answer & Explanation

Cynthia Herrera

Cynthia Herrera

Beginner2022-05-04Added 16 answers

It is true and, if the optimal is not on the boundary, it doesn'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 ) = 0
therefore,
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.

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