The simplex method is pretty much just a really advanced way to solve optimization problems using a more theoretical linear programming based approach rather than gradient descent.
@BryceSummers: That's the right idea---the simplex method is a method specifically for solving linear programs (LPs), i.e., minimize a linear objective subject to linear inequality constraints. Actually, the simplex method predates descent-based methods for LPs by quite some time. The main problem, though, was that the simplex method has (worst-case) exponential complexity. Later on, people discovered that you can solve LPs in polynomial time using interior point methods, which are (very roughly speaking) descent-based approaches that account for constraints. Still, the simplex method remains a common method of choice due to (often) good real-world performance on certain problems.