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102 CHAPTER 6. ITERATIVE EIGENVALUE SOLVERS
The Lagrange function is constructed as:
h(˜c, λ) =f(˜c)+λg(˜c). (6.57)
If ˜c is a minimum for the constrained problem (6.56), then there exists a real λ
such that (˜c, λ) is a stationary point for the Lagrange function (6.57). To compute
∂h
the stationary points, the first-order conditions for optimality are considered:
∂λ
and ∂h
∂˜c 1
,..., ∂h
∂˜c k
are computed. The partial derivatives with respect to ˜c i yield the
equations:
˜c i =
d iṽ i
d 2 for i =1,...,k. (6.58)
i + λ
These ˜c i , for i =1,...,k, can be substituted into the partial derivative of h(˜c, λ) with
respect to λ, which is:
∂h
∂λ = g(˜c) =˜c2 1 + ...+˜c 2 k − 1=0. (6.59)
This yields a univariate polynomial equation in the only unknown λ of degree 2k.
From this univariate polynomial the real-valued value for the Lagrange multiplier λ
can be solved. Once λ is known, the values for ˜c i are computed by (6.58) and are backtransformed
to c i by using the matrix V . In this way the solution c =(c 1 ,...,c k )
of problem (6.52) is computed. However, at this point this promising idea is not
yet implemented in combination with a Jacobi–Davidson method but is subject for
further research.
6.5 JDCOMM: A Jacobi–Davidson type method for commuting
matrices
This section introduces a Jacobi–Davidson eigenvalue solver for commuting matrices
to use in combination with the optimization method described in the previous sections
and chapters. For the specific application of polynomial optimization, this solver
is more efficient than other Jacobi–Davidson implementations. The optimization approach
described in the previous chapter has some promising properties which are used
in designing this efficient Jacobi–Davidson method: (i) because of the common eigenvectors,
all the matrices A pλ ,A x1 ,...,A xn commute, (ii) the matrices A x1 ,...,A xn
are much sparser than the matrix A pλ , and finally, (iii) only the smallest real eigenvalue
and its corresponding eigenvector of the matrix A pλ are required to locate the
(guaranteed) global optimum of p λ , without addressing any (possible) local optimum
p λ contains.
6.5.1 The method JDCOMM
Because the size of the commuting matrices mentioned in the previous section is
usually very large, N × N with N = m n and m =2d − 1, it is not advisable to
manipulate the full matrix in every iteration step of an algorithm and to address all