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When does \$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$ hold for a real function \$f(x,y)\$?

Let \$f(x,y)\$ be a real function of the variables \$x,y\$ (which can be real vectors). Under what conditions do we have the following equality: \$\$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$\$ For example, this equality is true if \$f(x,y) = xy\$ and \$x,y\$ are real scalars. Note that this is not the same as Von Neumann’s […]

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\$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$

Let \$f(x,y)\$ be a real function of the variables \$x,y\$ (which can be real vectors). Under what conditions do we have the following equality: \$\$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$\$ For example, this equality is true if \$f(x,y) = xy\$ and \$x,y\$ are real scalars. Note that this is not the same as Von Neumann’s […]

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Conditions for \$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$?

Let \$f(x,y)\$ be a real function of the variables \$x,y\$ (which can be real vectors). Under what conditions do we have the following equality: \$\$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$\$ For example, this equality is true if \$f(x,y) = xy\$ and \$x,y\$ are real scalars. Note that this is not the same as Von Neumann’s […]

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Conditions for \$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$?

Let \$f(x,y)\$ be a real function of the variables \$x,y\$ (which can be real vectors). Under what conditions do we have the following equality: \$\$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)\$\$ For example, this equality is true if \$f(x,y) = xy\$ and \$x,y\$ are real scalars. Note that this is not the same as Von Neumann’s […]