# Proving infeasibility using Duality

suppose we have the linear program min{$$c^Tx: Ax \leq 0, x \leq 0$$} and its corresponding dual

max{$$0^Tx: A^Ty \geq 0, y \leq 0$$}. How can we show that the Dual is infeasible? I started by contradiction and assumed the Dual is feasible, then its optimal value will be $$0$$ and by strong duality, the primal should also have an optimal value of $$0$$, however I am not able to reach a contradiction from this point.

The dual for this problem is $${\max g(\lambda_1,\lambda_2)\\\text{s. t.}\\\lambda_1,\lambda_2\succeq 0}$$where $$g(\lambda_1,\lambda_2){=\inf_{x}c^Tx+\lambda_1^TAx+\lambda_2^Tx\\=\inf_{x}(c+A^T\lambda_1+\lambda_2)^Tx}$$Now, when is the dual problem infeasible? How is it applied here?