A primal quadratic simplex algorithm tailored to the optimization over the vertices of a polytope is presented. Starting from a feasible vertex, it performs either strictly improving or admissible non-deteriorating steps in order to determine a locally optimum basic feasible solution in terms of the quadratic objective function. The algorithm so generalizes over local improvement methods for according applications, including in particular quadratic optimization problems whose feasible solutions correspond to vertices of a 0-1 polytope. Computational experiments for unconstrained binary quadratic programs, maximum cut, and the quadratic assignment problem serve as a proof of concept and underline the importance of a pivoting rule that is able to accept at least a restricted class of degenerate steps.