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A simple optimization problem is of form $\max_{x\in\mathcal R}f(x)$.

We can quantify as $\exists x\in\mathcal R\forall y\in\mathcal R f(y)\leq f(x)$.

The quantification here is $\exists\forall$.

  1. Is there a meaning to $\forall\exists$ from optimization?

  2. What is a good natural meaning to $\forall\exists$ quantification?

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    $\begingroup$ $\forall x\in\mathcal R.\exists y\in\mathcal R.f(y)\leq f(x)$ is simply true tautologically. $\endgroup$ – Derek Elkins Jul 17 '18 at 19:27
  • $\begingroup$ I am looking for reasonable analogous interpretation. $\endgroup$ – T.... Jul 17 '18 at 19:35
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The $\exists x \forall y$ form of quantification is used to indicate that there is some single object that has a particular relationship to all others, for instance an input that achieves the globally largest objective value in your example.

In $\forall y \exists x$ quantification, there can be a different $x$ for each $y$, but as pointed out in a comment, this is only interesting if the formula being quantified contains more than just a single inequality that allows setting $x=y$. The existence of locally optimal values (that fail to achieve a globally maximal objective value, but are perhaps only a short distance away and still achieve an absolute target value for the objective function) would be expressed in this way.

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  • $\begingroup$ Can you provide a concrete example? $\endgroup$ – T.... Jul 18 '18 at 1:59
  • $\begingroup$ Is it fair to tell optimization problems are in $\Sigma_2$? $\endgroup$ – T.... Jul 18 '18 at 4:37

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