I have a doubt regarding a constrained optimisation problem. Suppose my original constrained mi

rose2904ks 2022-06-20 Answered
I have a doubt regarding a constrained optimisation problem.

Suppose my original constrained minimisation problem is
min x f ( g ( x ) , x )  s.t.  g ( x ) = 3
I would like to know if this equivalent to solving the unconstrained minimisation problem
min x f ( 3 , x )
If not, when are these two problems equivalent?
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Answers (1)

lisicw2
Answered 2022-06-21 Author has 11 answers
The problems are certainly not equivalent in general. Take   f : R × R 4 R   to be given by
f ( y , x ) = y 2 + x 2   ,
and   g : R 4 R   to be given by
g ( x ) = 1 x   .
for instance. Then the minimum value of   f ( 3 , x )   is 9 , and occurs when   x = 0   , whereas
min x f ( g ( x ) , x )      s.t.        g ( x ) = 3
is   11 1 4  , and occurs when   x = 3 4 1  . I agree with Michal Adamszek's comment that the two problems will almost never be equivalent.
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