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50 CHAPTER 4. GLOBAL OPTIMIZATION OF MULTIVARIATE POLYNOMIALS
discarded because it is of no interest in this application.
The matrix A T p 1(x 1,x 2)
can be constructed using the same monomial basis B and
quotient space R[x 1 ,x 2 ]/I as introduced above. This yields the 9×9 matrix A T p :
1(x 1,x 2)
⎛
0 − 1 13
4 16
− 3
⎞
3
16 2
0 0 0 0
0 − 13
16
− 55
64
− 39
64
− 3 3
16 2
0 0 0
55
0
64
− 43 165
256 256
− 135
64
− 21
16
− 9 8
0 0
0 0 0 0 − 1 13
4 16
− 3 3
16 2
0
0 0 0 0 − 13
16
− 55
64
− 39
64
− 3 3
16 2
. (4.11)
27
55
0 32
0 0
64
− 43 165
256 256
− 135
64
− 21
16
9
0
64
− 9 8
0 0 0 0 − 1 13
4 16
⎜ 405 63 27
⎝
0
256 64 32
0 0 0 − 13
16
− 55 ⎟
64 ⎠
0 − 1503
1024
27
32
− 189
256
27
32
0 0
55
64
− 43
256
The real eigenvalues of the matrix A T p 1(x 1,x 2)
are computed as −0.402778, −0.201376,
0. The smallest one of these is the global minimum and its location can be read
off from the corresponding eigenvector. This (normalized) eigenvector is computed
as (1, −0.631899, 0.399297, 0.779656, −0.492664, 0.311314, 0.607864, −0.384108,
0.242718) T and has the same structure as the monomial basis B. The value of the x 1
coordinate can be read off from the eigenvector as it is the second element, −0.631899,
and the value of the x 2 coordinate can be read off as the fourth element, 0.779656.
This approach allows for computing the global minimum and its minimizer at once
without explicitly constructing the matrices A T x 1
and A T x 2
and without performing
eigenvalue/eigenvector computations on each of them.
As a final check, a local optimization technique, steepest descent, is applied to
the polynomial p 1 (x 1 ,x 2 ) from the starting locations (0.4, −0.6) and (−0.6, 0.8). The
first starting location results in a minimum at (0.3468, −0.6384) with criterion value
−0.2014. The second starting location results in a minimum at (−0.6319, 0.7797)
with criterion value −0.4028. The latter of these two values has the lowest criterion
value and matches the location and value found by both eigenvalue approaches.