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I know that Slater's condition implies strong duality and that the dual problem's supremum is attained. Is the infimum for the primal also attained under Slater's condition?

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It has no reason to be. For example, if you minimize $y$ subject to $y \ge e^x$ (that is, $e^x-y \le 0$) then the infimum is not attained; however, a point like $(0,2)$ shows that Slater's condition is satisfied.

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  • $\begingroup$ Thank you so much for the example! $\endgroup$
    – qp212223
    Nov 28, 2022 at 1:35

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