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48 CHAPTER 4. GLOBAL OPTIMIZATION OF MULTIVARIATE POLYNOMIALS
two local minima near (0.4, −0.6) T and (−0.6, 0.8) T , however it is unclear what their
exact location is and if one minimum has a lower value than the other.
Figure 4.2: Example polynomial p 1 (x 1 ,x 2 )
The first-order conditions for minimality of the polynomial p 1 (x 1 ,x 2 ) are given
by:
⎧
⎨
⎩
d (1) (x 1 ,x 2 ) = x 3 1 + 3 4 x2 1 +x 1 + 3 4 x 2 = 0
d (2) (x 1 ,x 2 ) = x 3 2 + 3 4 x 1 = 0
(4.9)
of which the real solutions constitute the stationary points of p 1 (x 1 ,x 2 ).
Note that this system of equations is indeed in Gröbner basis form already, with
respect to a total degree monomial ordering, since for both of the variables x 1 and x 2
there is an index i for which LT (d (i) ) is a pure power of the variable: LT (d (1) )=x 3 1
and LT (d (2) )=x 3 2. The ideal I is generated by d (1) (x 1 ,x 2 ) and d (2) (x 1 ,x 2 ) and
the quotient space R[x 1 ,x 2 ]/I is of dimension (2d − 1) n = 9. A basis is given by
the set of monomials {1,x 1 ,x 2 1,x 2 ,x 1 x 2 ,x 2 1x 2 ,x 2 2,x 1 x 2 2,x 2 1x 2 2}. In terms of this
basis (which corresponds to a convenient permutation of the total degree reversed
lexicographic monomial ordering) the matrices A T x 1
and A T x 2
are easily computed by