# Slater's Condition and Primal Optimal Value

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?

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.