Quadratic minimisation problems in statistics

• Published in: Journal of multivariate analysis Vol. 102; no. 3; pp. 698 - 713
• Main Authors: Albers, C.J., Critchley, F., Gower, J.C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.03.2011; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: Algebra; Canonical analysis; Canonical analysis Constraints Geometry Minimisation Quadratic forms Ratios Reduced rank; Constraints; Eigenvalues; Exact sciences and technology; Geometry; Mathematical functions; Mathematics; Minimisation; Multivariate analysis; Nonlinear algebraic and transcendental equations; Number theory; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Optimization; Probability and statistics; Quadratic forms; Quadratic programming; Ratios; Reduced rank; Sciences and techniques of general use; Statistics; Studies; Geometry; Minimisation; Reduced rank; 15A63; Canonical analysis; Ratios; Constraints; Quadratic forms; 15A21; 62H12; Transcendental equation; Eigenvector; Numerical linear algebra; Eigenvalue; Multivariate analysis; Non linear equation; Statistical method; Numerical analysis; Algebraic equation; Quadratic form
• ISSN: 0047-259X; 1095-7243; 1095-7243
• DOI: 10.1016/j.jmva.2009.12.018

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.