Consider the following problem:
$$\min_{x \in \mathbb{R}^n}f(x)=c^Tx$$ Subject to $ Ax=b$, where $A$ is full rank. Without any positive requirements (for instance, $x\ge0$), I want to show the following using Lagrange:
- If the problem has a bounded optimal solution, then all the feasible solutions are optimal
I guess the non-negative requirement would give a standard LP. Not having such requirement allows to solve the problem without incurring in infeasability problema due to to the negativity. My idea, up to now, is to use a lagrangian $L(x,\lambda)= c^Tx -\lambda(Ax-b)$ and try to derive some necessary and sufficient conditions for its optimallity.