# Intuitively, why is the complementary slackness condition true?

What's an intuitive proof that shows that the conditions of complementary slackness are indeed true:

1. If $x^*_j > 0$ then the $j$-th constraint in the dual is binding.
2. If the $j$-th constraint in the dual is not binding, then $x^*_j = 0$

And similarly for the dual variables $y^*_i$ and constraints in the Primal. Where $x^*$ and $y^*$ are the optimal solutions to the Primal and Dual respectively.

What's an intuitive proof as to why this is the case? Staring at the equations makes sense algebraically, but I wish to understand it at a more visceral level.

In order to understand what complementary slackness means, the concept of dual variables as "shadow prices" is useful. The dual variable associated with a primal constraint is called the constraint's shadow price because it can be thought of as how much the objective function would increase if the constraint was relaxed (meaning e.g. the right hand side of a $\le$ constraint was increased).
I find the geometric interpretation useful. Say we have the primal as $\max c x$ subject to $Ax \le b$ and $x \ge 0$. We know that optimum solutions are vertices of the polytope defined by the constraints. Each such vertex is defined by the intersection of $n$ linearly independent hyperplanes defined by the constraints. When is a vertex solution $x^*$ optimal for the direction $c$? It is optimal iff the vector $c$ is in the cone of the rows of $A$ defining the vertex $x^*$ (that is the vector $c$ can be written as a non-negative combination of the rows defining $x^*$). Otherwise we can improve the solution. The dual variables corresponding to the rows defining $x^*$ are strictly positive and the rest are $0$. This shows dual complementary slackness. Once we have strong duality we also get primal complementary slackness.