Motivation non-negativity constraint KKT conditions

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I am having problems understanding the non-negativity constraint on the Lagrange multiplier in the KKT conditions. Why does the sign matter?

See this link for statement of the KKT conditions.

I do understand that it's about dual feasibility, but why is dual feasibility defined this way?

Is there an intuitive way to explain this constraint on $\mu\geq0$?

Thank you!