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The performance of two different approaches to solving the non-linear control allocation problem is presented. The non-linear control allocation problem is formulated using piecewise linear functions to approximate the control moments produced by a set of control effectors. when the control allocation problem is formulated as a piecewise linear program, an additional set of constraints enter into the problem formation. One approach is to introduce a set of binary variables to enforce these constraints. The result is a mixed- integer linear programming problem that can be solved using any branch-and-bound software. A second approach is to solve the piecewise linear programming problem using a modified simplex method. The simplex algorithm is modified to enforce a subset of the decision variables to enter into the basis only if certain conditions are met. We will show that solving the optimization problem using the simplex based approach is significantly faster than solving the same problem using a mixed-integer formulation. We will then compare the closed-loop performance of a re-entry vehicle using both approaches.