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Take a generic LP, say we wish to maximize $\mathbf{c}^{\top} \mathbf{x}$ subject to $A \mathbf{x} = \mathbf{b}$.

In the standard partitioning into basics/nonbasics, we know that $\mathbf{x} = \begin{bmatrix} B^{-1} \mathbf{b} \\ \mathbf{0} \end{bmatrix}$ is a solution to $A \mathbf{x} = \mathbf{b}$.

So this is a 'particular solution'. But what is the general solution form then? Say all solutions are of the form $$\begin{bmatrix} B^{-1} \mathbf{b} \\ \mathbf{0} \end{bmatrix} + M \mathbf{\alpha}$$ for some $\mathbf{\alpha} \in \mathbb{R}^n$. Is there a closed form expression for $M$? How do we work it out?

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    The set of optimal solutions is a polytope, so any expression that defines a polytope can do the job. – 2017-02-07

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