Active set methods differ from the simplex method for linear programming in that neither the iterates nor the solution need be vertices of the feasible set. When the quadratic programming problem is nonconvex, these methods usually find a local minimizer. Finding a global minimizer is a more difficult task that is not addressed by the software currently available. Equality-constrained quadratic programs arise, not only as subproblems in solving the general problem, but also in structural analysis and other areas of application. This matrix can be computed with orthogonal factorizations or, in the case of sparse problems, by factorization of a submatrix of , just as in the simplex method for linear programming.
Most codes uses null-space methods.
Although this approach works only for a subclass of problems, there are many applications in which it is useful. Given a feasible , these methods find a direction by solving the subproblem where q is the objective function and is a working set of constraints. In all cases is a subset of the set of constraints that are active at . Typically, either is equal to or else has one fewer index than . The working set is updated at each iteration with the aim of determining the set of active constraints at a solution .