§ Linear optimisation is the same as linear feasibility checking

Core building block of effectively using the ellipsoid algorithm.

• If we posess a way to check if a point $p \in P$ where $P$ is a polytope, we can use this to solve optimisation problems.
• Given the optimisation problem maximise $c^Tx$ subject to $Ax = b$, we can construct a new non-emptiness problem. This allows us to convert optimisation into feasibility .
• The new problem is $Ax = b, A^Ty = c, c^Tx = b^T y$. Note that by duality, a point in this new polyhedra will be an optimal solution to the above linear program . We are forcing $c^Tx = b^Ty$, which will be the optimal solution, since the solution where the primal and dual agree is the optimal solution by strong duality.
• This way, we have converted a linear programming problem into a check if this polytope is empty problem!