Quadratic minimisation problems in statistics

Albers, C.J.; Critchley, F. and Gower, J. C. (2011). Quadratic minimisation problems in statistics. Journal of Multivariate Analysis, 102(3) pp. 698–713.

DOI: https://doi.org/10.1016/j.jmva.2009.12.018

Abstract

We consider the problem min_x(x – t)’A(x – t) subject to x’Bx + 2b’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.

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