Quadratic minimisation problems in statistics

We consider the problem minx(x−t)′A(x−t) subject to x′Bx+2b′x = k where A is positive definite or positive semidefinite. Commonly occurring statistical variants of this problem are discussed within the framework of a general unifying methodology. These include non-trivial considerations that arise when (i) A and/or B are not of full rank and (ii) t takes special forms (especially t = 0 which, under further conditions, reduces to the well-known two-sided eigenvalue solution). Special emphasis is placed on insights provided by geometrical interpretations. Algorithmic considerations are discussed and examples given.