How to prove the existence of the saddle point implies the below equations? p ∗ = d ∗ = L ( v 0 , x 0 )

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How to prove the existence of the saddle point implies the below equations?      p    ∗    =      d    ∗    =  L  (      v    0    ,      x    0    )

Kroatujon3 · Accepted Answer

Step 1Suppose   (      v    0    ,      x    0    ) is a saddle point. Note that                    ∀        x        ,        L        (                  v          0                ,        x        )        ≤        L        (                  v          0                ,                  x          0                )                                    ⟹                          sup          x                L        (                  v          0                ,        x        )        =        L        (                  v          0                ,                  x          0                )                                                  ⟹                          inf          v                          sup          x                L        (        v        ,        x        )        ≤        L        (                  v          0                ,                  x          0                )        .             .Suppose the inequality is strict. Then there exists some       v    1   such that      sup    x    L  (      v    1    ,  x  )  &amp;lt;  L  (      v    0    ,      x    0    )  . .But then,  L  (      v    0    ,      x    0    )  ≤  L  (      v    1    ,      x    0    )  ≤      sup    x    L  (      v    1    ,  x  )  &amp;lt;  L  (      v    0    ,      x    0    )  . .This is a contradiction, hence       p    ∗    =  L  (      v    0    ,      x    0    ) .We can show the other equality similarly, or simply by replacing L with -L, and swapping the roles of v and x.

How to prove the existence of the saddle point implies the below equations? p &#x221

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