Quadratic minimisation problems in statistics

We consider the problem min x ( x − t ) ′ A ( x − t ) subject to x ′ B x + 2 b ′ x = k where A  is positive definite or positive semi-definite. 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.