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Applications of the alternating direction method of multipliers to the semidefinite inverse quadratic eigenvalue problem with a partial eigenstructure. (English) Zbl 1280.65036

The semidefinite inverse quadratic eigenvalue problem (SDIQEP) is defined as follows. Given the solution \((X,\Lambda)\) to the symmetric positive semidefinite quadratic eigenvalue problem, i.e., \(MX\Lambda^2+CX\Lambda+KX=0\), find the matrices \((M,C,K)\) all in \(\mathbb{R}^{n\times n}\). Now suppose that \((X,\Lambda)\) is partially known (only \(p\leq n\) eigenpairs) for the symmetric matrices \((M_a,C_a,K_a)\) and one wants to find the symmetric matrices \((M,C,K)\) with \(M\) and \(K\) positive semidefinite that have these prescribed eigenpairs and that are as close as possible (in Frobenius norm) to the triple \((M_a,C_a,K_a)\). This is a constrained least squares problem, and hence can be reformulated as a projection problem, that has to be solved iteratively. The problem is to maintain semidefiniteness and sparsity during the iterations. It is proposed to use the alternating direction method of multipliers (ADMM) of R. Glowinski and A. Marroco [Rev. Franc. Automat. Inform. Rech. Operat. 9, Analyse numer., No. R–2, 41–76 (1975; Zbl 0368.65053)]. Three variants of the algorithm applied to this problem are given and a convergence analysis is given. Several numerical experiments verify the effectiveness of the methods and the sensitivity to the choice of some of the parameters in the methods.

MSC:

65F18 Numerical solutions to inverse eigenvalue problems
15A29 Inverse problems in linear algebra
15A18 Eigenvalues, singular values, and eigenvectors

Citations:

Zbl 0368.65053
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