Determining if a linear program is self-dual?

Question

I want to show that if $A$ is a skew-symmetric $n\times n$ matrix ($A^{T}=-A$ ) and $c\in \mathbb{R}^{n},$ then $$\begin{array}{cc} \max & c^{T}x \\ s.t. & Ax\geq c \\ & x\geq 0 \end{array} $$ is self-dual.

I started my solution by changing the objective: $\max c^{T}x $ to $\min -c^{T}x $.

I then obtain the primal program as

$$ \begin{array}{cc} \min & -c^Tx \\ s.t. & Ax \geq c \\ & x \geq 0 \\ \end{array} $$

and the corresponding Dual program as

$$ \begin{array}{cc} \max & y^Tc \\ s.t. & A^Ty \leq -c \\ & y \geq 0 \\ \end{array} $$

I know that if a linear program is self-dual then the primal is the same as the dual. My question is how do I establish this?

Answer 1

In solving linear programming(LP) by innter pointer method, we can build a equivalent "homogeneous and self-dual linear feasibility optimization problem" (HLF) for any LP. Then, we can get the optimal solution of original LP through solving the HLF.

I've been learning about this recently and would love to discuss it.

The references are: Andersen, E. D., & Andersen, K. D. (2000). The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm. In High performance optimization (pp. 197-232). Boston, MA: Springer US.

Xu, X., Hung, P. F., & Ye, Y. (1996). A simplified homogeneous and self-dual linear programming algorithm and its implementation. Annals of Operations Research, 62(1), 151-171.