A Dual Optimization Approach to Inverse Quadratic Eigenvalue Problems with Partial Eigenstructure

Abstract

The inverse quadratic eigenvalue problem (IQEP) arises in the field of structural dynamics. It aims to find three symmetric matrices, known as the mass, the damping, and the stiffness matrices, respectively, such that they are closest to the given analytical matrices and satisfy the measured data. The difficulty of this problem lies in the fact that in applications the mass matrix should be positive definite and the stiffness matrix positive semidefinite. Based on an equivalent dual optimization version of the IQEP, we present a quadratically convergent Newton-type method. Our numerical experiments confirm the high efficiency of the proposed method.

Correlation stress testing for value-at-risk: an unconstrained convex optimization approach

“…3 The shrinkage method of Kupiec [17] and the sequential single-stress method of Turkey, Epperlein, and Christofides [34], where the case C 1 = C 1 and C 2 = C 2 is formally referred to as the local correlation stress testing, both are capable of handling the constraints (H1)-(H3), but, as commented by Rebonato and Jäckel [25] that "there is no way of determining to what extent the resulting matrix is optimal in any easily quantifiable sense". 4 Finger's method as well as other spectral decomposition based methods proposed in those studies also suffer similar drawbacks.…”

“…But, since X is a symmetric matrix, the lower part of fixed elements is automatically included. Our eventual goal is to find the nearest correlation matrix to C from all those of satisfying conditions in (4). This leads to the following least-square optimization problem:…”

Applications of the alternating direction method of multipliers to the semidefinite inverse quadratic eigenvalue problem with a partial eigenstructure

“…As other efficient applications of ADMM, the main advantage of ADMM for SDIQEP is that the resulting subproblems are easy (some are easy enough to have closed-form solution while the others are standard minimization problems which can be easily solved up to high precisions by existing methods). We shall compare these ADMM schemes numerically with some existing methods including the interior-point approach mentioned in [25] and the generalized Newton approach in [1], for some large scale cases of SDIQEP.…”

“…This difficulty of dimensionality excludes efficient applications of interior-point or Newton-like algorithms for (4), for which solving a system of equations whose dimensionality is proportional to np is unavoidable at each iteration. For existing algorithms for SDIQEP, we refer to [24] for an algorithm for solving a relaxed version of SDIQEP where the semidefiniteness constraints on M and K in (3) are removed from consideration; [1] for a semismooth generalized Newton method which is capable of solving large-scale cases, and [25] for some semidefinite programming techniques, which may not be so effective for handling large-scale problems.…”

“…Aiming at generating a updated pencil P (λ) := λ 2 M + λC + K such that it coincides with the measured partial eigendata (X, Λ) while is closest to the original pencil (1) in sense of least squares, the updated matrices (M, C, K) via solving SDIQEP obviously are required to preserve structural properties of the analytic matrices M a , C a , K a . In this paper, we mainly focus on the semidefiniteness requirement (see for instance the applications mentioned in [7,10]).…”