Least-Squares Covariance Matrix Adjustment

We consider the problem of finding the smallest adjustment to a given symmetric n × 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 around n = 45. In this paper we formulate a dual problem that has no matrix inequality or 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 sheds light on the least-squares covariance adjustment problem, and in many situations 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. When structure in the problem (such as sparsity or low rank) is exploited, far larger problems can be solved.