# What is the relation between the Lagrange multipliers and the solution of the dual in a linear optimisation problem?

Consider the following linear minimisation problem
$$(1) quad min_{xgeq 0} a^top x,\ quad quad quad text{s.t. }B^top x=c$$
where

• $$x$$ is the $$ptimes 1$$ vector of unknowns.

• $$a$$ is a $$ptimes 1$$ vector of real numbers.

• $$B$$ is a $$ptimes k$$ matrix of real scalars.

• $$c$$ is a $$ktimes 1$$ vector of real scalars.

• $$p>k$$.

Consider the Lagrangian of (1)

$$(2) quad L(x,mu,nu)= a^top x+mu^{top}left(B^top x-cright)+nu^{top}x,$$

Consider also the dual of (1)
$$(3) quad max_{y} c^top y,\ quad quad quad text{s.t. } By leq a$$

Question: what is the relation between the dual of (1) and the Lagrange multipliers of (1)?

Let $$mathcal{X}^*$$ be the set of argmin of (1).

Let $$mathcal{Y}^*$$ be the set of argmax of (3).

Let $$mathcal{M}^*$$ be the set of Lagrange multipliers $$mu$$ corresponding to each element of $$mathcal{X}^*$$.

Let $$mathcal{V}^*$$ be the set of Lagrange multipliers $$nu$$ corresponding to each element of $$mathcal{X}^*$$.

Is $$mathcal{Y}^*=mathcal{M}^*$$? What is the relation between $$mathcal{Y}^*$$ and $$mathcal{V}^*$$?