## § 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, wecan use this to solve optimisation problems.
• Given the optimisation problem maximise $c^Tx$ subject to $Ax = b$, we canconstruct a new non-emptiness problem. This allows us to convert optimisationinto 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 thesolution where the primal and dual agree is the optimal solution by strongduality.
• This way, we have converted a linear programming problem into acheck if this polytope is empty problem!