Im looking at this question about duality in parametric problems:
And the proposed solution:
But I'm having trouble understanding what the solution means from the part where it starts talking about $\bar{\mu}$ being an optimal solution for the given problem. Why does the inequality hold?
For the optimization problem $$ \min_x f(x) \quad \text{ s.t. } \quad x \in X,\, g(x) \leq \bar{u} $$ the Lagrangian Dual Problem is (as also explained in the provided solution) $$ \max_\mu \left(\inf_{x \in X} f(x) + \mu^\top (g(x) - \bar{u})\right) \quad \text{ s.t. } \quad \mu \geq 0_r. $$ The Lagrange multiplier $\bar{\mu}$ (of the original problem) is the optimal solution of this dual problem and hence for all $\mu \geq 0_r$ it holds that $$ \inf_{x \in X} f(x) + \bar{\mu}^\top (g(x) - \bar{u}) \geq \inf_{x \in X} f(x) + \mu^\top (g(x) - \bar{u}). $$In particular, this holds for $\mu = \tilde{\mu}$ and this is the argument used in the inequality in your provided solution.