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In a linear programming context, does the primal optimal solution yield an explicit way to find the primal dual solution? I vaguely remember something like this from an optimization class but can't recall the precise statement.

The closest I've found is that if $x, y$ are primal and dual feasible respectively, and the value of the primal objective at $x$ is equal to the value of the dual objective at $y$, then $x$ and $y$ are primal and dual optimal.

Thanks!

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Say you have an optimal solution to the primal. You now wish to find an optimal solution to the dual. If such a dual solution exists, then it must satisfy complementary slackness by strong duality. You can employ this to find optimal dual solution(s).

See the Complementary Slackness section of the wiki article on Linear Programming: https://en.wikipedia.org/wiki/Linear_programming

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