Semi-definite programming techniques for structured quadratic inverse eigenvalue problems

Abstract

In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs.

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

“…More specifically, as in [25], we can reformulate the SDIQEP model (4) as a standard semidefinite programming (SDP) problem defined as follows:…”

“…For Algorithms 1-3, "t 1 " and "t 2 " are the time of computing Π Ω + and Π S B , respectively. Moreover, for Algorithm 2, "t 3 " is the aggregated time of computing the step size θ k defined in (25). Since both the compared approaches are applicable to large scale cases of SDIQEP, we observe their difference numerically for more scenarios of n and p than Section 5.1.1, and we test the sensitivity of their difference to different values of n and p. First, in Table 3, we fix p = 30 for Example 5.1 and compare these two approaches for different values of n. We see that the ADMM approach is more efficient than the generalized Newton approach, as it requires much less computing time.…”

“…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.…”

Semidefinite inverse eigenvalue problems with prescribed entries and partial eigendata

“…Without causing any confusion, we regard the minimization problem (1) as the PESDIEP. As noted in [43], one may find a solution to the PESDIEP (1) by using classical semidefinite programming (SDP) techniques (see for instance [1,2,53]). However, the primal-dual interiorpoint methods may not be effective for solving large-scale semidefinite programming problems [53].…”

“…We also extend the proposed method to the case of lower bounds. We report some numerical tests, including the comparison with the interior-point approach mentioned in [43,53] for solving the PESDIEP, to illustrate the effectiveness of our method.…”