Let's consider following problem:
$$\text{minimize}\ \ \ \ f(x)\\\text{subject to}\ \ x\in X , \ Ax = b$$
Then I'd like to prove following claims:
1) If $f$ is finite and that there exists $\bar x \in ri(X)$ such that $A\bar x = b$. Then $f^* = q^*$ and there exist at least one dual optimal solution.
2) There holds $f^* = q^*$ and $(x^*, \lambda^*)$ are a primal and dual optimal solution pair iff if $x^*$ is feasible and $x^* \in argmin_{x\in X}L(x,\lambda^*)$
Any advice hint to start the proof ?