So I know that in general the simplex method for linear and convex quadratic programming can require exponential time. But assuming a positive semidefinite quadratic program that is solvable by the simplex method, then the problem is definitely in P, right?
The reason I ask: I am working on getting a handle on complexity theory. And I think I found a reduction from a problem to a positive semidefinite quadratic problem BUT the solution is only valid if the returned optimum is at a vertex of the feasible region. So that leaves out solving it with an interior point method since the minimum can be trivially obtained at the center of the feasible region and maybe at other points too. Since the simplex method operates on vertices, then I know that once the optimum is found, then it is at a point I actually care about.
Assuming the reduction is proper and always results in a correct answer: is reduction to the simplex method a sufficient condition for proving membership in P? OR does someone have a good reference to efficient exterior point methods for this class of problem? Is there another method for quadratic programs that I am overlooking? OR am I basically stuck with proving that the number of iterations is polynomial for this specific program? Is this question better suited to a different stack exchange site?