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118 CHAPTER 7. NUMERICAL EXPERIMENTS
Table 7.10: Sparsity of the matrices A p1
and A x1 ,A x2 ,A x3 ,A x4
Matrix of dimension 2401 × 2401 Number of non-zeros % filled
A p1 1326556 23.01
A x1 63186 1.10
A x2 52558 0.91
A x3 52663 0.91
A x4 54238 0.94
To put the performance of this method into perspective, its performance is compared
with the performance of the JD method. The JD method iterates with the relatively
‘expensive’ operator A pλ , whereas the JDCOMM method tries to outperform the JD
method by targeting on the smallest real eigenvalue while iterating with a ‘cheap’
operator A xi in the inner loop and with the operator A pλ in the outer loop. Finally,
the outcomes of both the methods JD and JDCOMM are compared to the outcomes of
the SOSTOOLS and GloptiPoly software.
All the results throughout this section are obtained with Matlab version R2007B
running on an Intel Pentium PIV 2.8GHz platform with 1024MB of internal memory.
7.5.1 Experiment 1
Here we demonstrate the computation of the global minimum of a multivariate polynomial
by using the JD and the JDCOMM method. For the first experiment a polynomial
p λ with n =4,d =4,m = 7, and λ = 1 is considered:
p 1 (x 1 ,x 2 ,x 3 ,x 4 )=(x 8 1 + x 8 2 + x 8 3 + x 8 4)+
14x 4 1x 2 x 3 x 4 +6x 1 x 4 2x 3 x 4 − 11x 3 2x 2 3x 2 4 + x 1 x 2 x 2 3x 2 4 +8x 2 x 3 x 3 4+
+x 3 x 2 4 +3x 1 x 2 + x 2 x 3 +2x 3 x 4 + x 4 +8.
(7.6)
Considering the quotient space R[x 1 ,x 2 ,x 3 ,x 4 ]/I of dimension N =(2d − 1) n =
2401, the matrices A p1 , A x1 , A x2 , A x3 , and A x4 are constructed explicitly. From the
2401 eigenvalues of these matrices, we are interested in the smallest real eigenvalue
and corresponding eigenvector of the matrix A p1 . Table 7.10 shows the differences in
the number of non-zero elements of all the involved matrices. The matrices A xi , i =
1,...,4, are much sparser than the matrix A p1 . See also Figure 7.3 for a representation
of the sparsity structure of these matrices.
One approach would be to compute all the eigenvalues of the matrix A p1 using a
direct solver and to select the smallest real one as being the global optimum of polynomial
p 1 (x 1 ,x 2 ,x 3 ,x 4 ). A similar approach would be to compute all the eigenvalues
of the matrices A x1 , A x2 , A x3 or A x4 and to read off the coordinates of the stationary
points from the corresponding eigenvectors. The global optimum of p 1 (x 1 ,x 2 ,x 3 ,x 4 )
can then be selected from all the stationary points by computing the associated function
values and to pick out the smallest real one. Computing all the eigenvalues of the
matrices A p1 , A x1 , A x2 , A x3 , and A x4 using a direct solver takes 57.9, 56.4, 54.5, 52.8,
and 48.8 seconds respectively. The global minimizer we are looking for has the value