We are always told that the Simplex Algorithm is meant for solving linear equations with linear constraints.
But how badly would the Simplex Algorithm perform if we implemented it on a typical non-linear problem that is better suited for algorithms such as the Interior Point Method?
For instance, the Simplex Algorithm "scans" different vertices on the exterior surface made by the intersection of all equations and constraints - I am not sure if the surface made by non-linear equations and non-linear constraints would even have vertexes?
For argument sake, if someone insisted on using the Simplex Algorithm on a non-linear problem, would this fail instantly? Or could it still in theory return an acceptable answer, but it would be unlikely that this answer was the true optimal?