Totally singular quadratic minimization problems

A survey of necessary, sufficient, and necessary and sufficient conditions for optimality for a class of time-varying singular quadratic minimization problems is presented. The development of the theory is traced from the time of discovery of the generalized Legendre-Clebsch condition. The most important conditions are stated in the form of theorems and are proved. Certain lengthy proofs are only outlined and the reader is referred to the literature for details. The reference list is not intended to be a complete bibliography; rather, it contains, in this author´s opinion, the most important references in this area.