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If I've been given a problem that tells me that a certain LP1 is a primal, and it's dual is LP2.. Could I also say that LP2 is the primal, and LP1 is it's dual?

Why do we call one a primal and one a dual?

The weak duality theorem states that the objective function value of the dual at any feasible solution is always greater than or equal to the objective function value of the primal at any feasible solution.

If the names are interchangeable, does that mean that the objective function value of the LP1 at any feasible solution is >= to the objective function value of the LP2 at any feasible solution? Despite LP1 being the "primal" and LP2 it's dual?

Eric Nelson
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1 Answers1

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The dual of the dual of an LP is the same LP again, so LP2 is the dual of LP1 iff LP1 is the dual of LP2. Which one you choose to call the primal is a matter of where you're choosing to place your attention: it's not an intrinsic fact about the LPs themselves.

Analogously, given, say, a complex number $z$, you can consider its complex conjugate $\overline{z}$. The complex conjugate of the complex conjugate of $z$ is equal to $z$ again, so it's equally valid to start with $\bar{z}$ and consider its complex conjugate $z$.

Qiaochu Yuan
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