Show that f ( x 1 ∗ , . . . x n ∗ ) = max { f ( x 1 , . . . , x n ) : ( x 1 , . . . , x n ) ∈ Ω } if and only if − f ( x 1 ∗ , . . . x n ∗ ) = min { − f ( x 1 , . . . , x n ) : ( x 1 , . . . , x n ) ∈ Ω }I am not exactly sure how to approach this problem -- it is very general, so I can't assume anything about the shape of f. It seems obvious that flipping the max problem with a negative turns it into a min problem. Thoughts?

Question

Show that   f  (      x    1    ∗    ,  .  .  .      x    n    ∗    )  =  max  {  f  (      x    1    ,  .  .  .  ,      x    n    )  :  (      x    1    ,  .  .  .  ,      x    n    )  ∈  Ω  } if and only if   −  f  (      x    1    ∗    ,  .  .  .      x    n    ∗    )  =  min  {  −  f  (      x    1    ,  .  .  .  ,      x    n    )  :  (      x    1    ,  .  .  .  ,      x    n    )  ∈  Ω  }I am not exactly sure how to approach this problem -- it is very general, so I can&#039;t assume anything about the shape of   f. It seems obvious that flipping the max problem with a negative turns it into a min problem. Thoughts?

allstylekvsvi · Accepted Answer

Follow the chain of equivalences here:                          f        (                  x          1          ∗                ,        .        .        .        ,                  x          n          ∗                )        =        max        {        f        (                  x          1                ,        .        .        .        ,                  x          n                )        :        (                  x          1                ,        .        .        .        ,                  x          n                )        ∈        Ω        }                                    ⟺                                    (                  x          1          ∗                ,        .        .        .        ,                  x          n          ∗                )        ∈        Ω         and         ∀        (                  x          1                ,        .        .        .        ,                  x          n                )        ∈        Ω        ,        f        (                  x          1          ∗                ,        .        .        .                  x          n          ∗                )        ≥        f        (                  x          1                ,        .        .        .        ,                  x          n                )                                    ⟺                                    (                  x          1          ∗                ,        .        .        .        ,                  x          n          ∗                )        ∈        Ω         and         ∀        (                  x          1                ,        .        .        .        ,                  x          n                )        ∈        Ω        ,        −        f        (                  x          1          ∗                ,        .        .        .        ,                  x          n          ∗                )        ≤        −        f        (                  x          1                ,        .        .        .        ,                  x          n                )                                    ⟺                                    −        f        (                  x          1          ∗                ,        .        .        .        ,                  x          n          ∗                )        =        min        {        −        f        (                  x          1                ,        .        .        .        ,                  x          n                )        :        (                  x          1                ,        .        .        .        ,                  x          n                )        ∈        Ω        }        .

Show that f ( <msubsup> x 1 ∗ </msubsup> , . .

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