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126 CHAPTER 7. NUMERICAL EXPERIMENTS
Table 7.16: Sparsity of the matrices A p1
and A x1 ,A x2 ,A x3 ,A x4
Matrix of dimension 6561 × 6561 Number of non-zeros %filled
A p1 1255231 2.92
A x1 45717 0.11
A x2 59756 0.14
A x3 41129 0.10
A x4 41712 0.10
real eigenvalue immediately, but then the JDCOMM methods become (almost) just as
fast as the conventional JD method. Respectively, the JD and JDCOMM method with
the matrices A x1 , A x2 , A x3 , A x4 , and A x5 need 3.5, 3.7, 3.5, 3.8, 3.7, and 3.6 seconds
and 2.1×10 8 ,2.2×10 8 ,2.2×10 8 ,2.2×10 8 ,2.2×10 8 , and 2.2×10 8 flops. To compute
the same global minimizer including its location, SOSTOOLS and GloptiPoly require
as much as 9.2 and 8.7 seconds.
7.5.4 Experiment 4
In this final experiment we study an optimization problem of the same dimensions as
the previous one: n =4,d = 5, and β = 1 with the following Minkowski-dominated
polynomial:
p 1 (x 1 ,x 2 ,x 3 ,x 4 )=(x 10
1 + x 10
2 + x 10
3 + x 10
4 )+3x 5 1x 2 x 3 x 4 +
+x 5 1x 3 x 2 4 +7x 5 1x 3 x 4 − 5x 3 1x 2 3x 4 +
−7x 2 1x 2 x 2 3x 2 4 − 5x 1 x 3 3x 3 4 − 5.
(7.9)
The involved quotient space and thus also the matrices A p1 , A x1 , A x2 , A x3 , and
A x4 are of dimension 6561 × 6561. Table 7.16 shows the differences in the number of
nonzero elements between those matrices. See Figure 7.10 for a representation of the
sparsity structure of these matrices.
The global minimizer, the smallest real eigenvalue of the matrix A p1 , we are
looking for has value −206.5 and is located at the point with coordinates x 1 =1.79,
x 2 =1.43, x 3 = −1.54, and x 4 =1.54, which are the corresponding eigenvalues of the
matrices A x1 , A x2 , A x3 , and A x4 . The location of these eigenvalues with respect to
the entire eigenvalue spectrum of each matrix is depicted with red dots in Figure 7.11.
The smallest real eigenvalue of the matrix A p1 is not located very close to the exterior
of the spectrum: there are 34 eigenvalues with a real part smaller than −206.5.
See the top-left subfigure of Figure 7.11.
The smallest real eigenvalue is computed with the JD and JDCOMM method using
various search space parameters. The minimal (mindim) and maximal (maxdim)
dimensions of the search space are varied between 5 and 100. See Table 7.17 for
the results. Blanks in this table indicate that the corresponding method with these
settings is unable to converge to the requested eigenvalue using a maximum of 1000
outer iterations and 10 inner iterations with GMRES. Note here that the standard JD