In this work we examine the problem of minimization of a positive semidefinite quadratic form under linear constraints. We introduce the use of various kinds of outer generalized inverses, define such outer inverses depending on the constraint set and present their properties. In addition, the minimizing vector of this problem has an additional property: it should belong to the set perpendicular to the kernel of the quadratic form examined. Finally, the T-restricted outer inverse of a matrix is introduced, covering all the cases presented in several published papers.