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Least-squares covariance matrix adjustment. (English) Zbl 1099.65039
SUMMARY: We consider the problem of finding the smallest adjustments to a given symmetric $$n\times n$$ matrix, as measured by the Euclidean or Frobenius norm, so that it satisfies some given linear equalities and inequalities, and in addition is positive semidefinite. This least-squares covariance adjustment problem is a convex optimization problem, and can be efficiently solved using standard methods when the number of variables (i.e., entries in the matrix) is modest, say, under 1000. Since the number of variables is $$n(n+1)/2,$$ this corresponds to a limit $$n=45.$$ J. Malick [SIAM J. Matrix Anal. Appl. 26, 272–284 (2004; Zbl 1080.65027)] studies a closely related problem and calls it the semidefinite least-squares problem.
In this paper we formulate a dual problem that has no matrix inequality or matrix variables, and a number of (scalar) variables equal to the number of equality and inequality constraints in the original least-squares covariance adjustment problem. This dual problem allows us to solve far larger least-squares covariance adjustment problems than would be possible using standard methods.
Assuming a modest number of constraints, problems with $$n=1000$$ are readily solved by the dual method. The dual method coincides with the dual method proposed by Malick [loc. cit.] when there are no inequality constraints and can be obtained as an extension of his dual method when there are inequality constraints. Using the dual problem, we show that in many cases the optimal solution is a low rank update of the original matrix. When the original matrix has structure, such as sparsity, this observation allows us to solve very large least-squares covariance adjustment problems.

##### MSC:
 65F30 Other matrix algorithms (MSC2010) 90C06 Large-scale problems in mathematical programming 90C22 Semidefinite programming 90C46 Optimality conditions and duality in mathematical programming 65F20 Numerical solutions to overdetermined systems, pseudoinverses 15B57 Hermitian, skew-Hermitian, and related matrices
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