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Chapter 6
Iterative eigenvalue solvers in
polynomial optimization
Because only the smallest real eigenvalue of the large and sparse matrix A pλ is needed
for the polynomial optimization problem of a Minkowski dominated polynomial, it
is a plausible choice to compute this eigenvalue with an iterative eigenvalue solver.
Iterative eigenvalue solvers have the useful feature that they can focus on a certain
subset of the eigenvalue spectrum. Moreover they can be implemented to operate in
a matrix-free fashion: iterative solvers accept a device which computes the matrixvector
product on a given vector as input. This avoids the explicit construction of the
matrix under consideration and enables the use of the nD-systems approach as described
in Chapter 5. Iterative methods are based, for example, on an Arnoldi method
or on a Jacobi–Davidson method. The Jacobi–Davidson (JD) method as presented by
Sleijpen and van der Vorst (see [94]) is among the best iterative eigenvalue solvers. In
the literature its implementation is called the JDQR method, for standard eigenvalue
problems, and the JDQZ method, for generalized eigenvalue problems.
In the first two sections of this chapter a general introduction to iterative eigenvalue
solvers is given, followed by a brief explanation of the techniques used in Jacobi–
Davidson solvers (based on [58]). In the Sections 6.3 and 6.4 some of our own modifications
of the Jacobi–Davidson method, which aim to improve its efficiency for
polynomial optimization, are introduced. Among these improvements are the ability
to focus on real eigenvalues only, which is non-standard among iterative solvers, and
a procedure to project an approximate eigenvector to a close-by vector with Stetter
structure. The development of a Jacobi–Davidson eigenvalue solver for a set of commuting
matrices is described in Section 6.5 of this chapter. This Jacobi–Davidson
method is called the JDCOMM method and has two distinctive properties: (i) it is able
to target the smallest real eigenvalue of a large sparse matrix and moreover (ii) it
computes the eigenvalues of the sparse matrix A pλ while iterating with one of the
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