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44 CHAPTER 4. GLOBAL OPTIMIZATION OF MULTIVARIATE POLYNOMIALS
4.1 The global minimum of a Minkowski dominated polynomial
Let q(x 1 ,...,x n ) ∈ R[x 1 ,...,x n ] be a real polynomial. We are interested to compute
its infimum over R n . Let d be a positive integer such that 2d (strictly) exceeds the
total degree of q(x 1 ,...,x n ) and consider the one-parameter family of what will be
called (Minkowski) dominated polynomials:
p λ (x 1 ,...,x n ):=λ(x 2d
1 + ...+ x 2d
n )+q(x 1 ,...,x n ), λ ∈ R + . (4.1)
Note that the nomenclature for this family derives from the property that the
value of p λ is dominated by the term λ(x 2d
1 + ...+ x 2d
n ) when the Minkowski 2d-norm
‖(x 1 ,...,x n )‖ 2d =(x 2d
1 + x 2d
2 + ... + x 2d
n ) 1/(2d) becomes large. Consequently, the
polynomial p λ has a global minimum over R n for each λ ∈ R + : the presence of the
dominating term λ(x 2d
1 + ...+ x 2d
n ) ensures that p λ has a global minimum. In fact
information about the infimum (and its ‘location’) of an arbitrary real polynomial
q(x 1 ,...,x n ) can be obtained by studying what happens to the global minima and
the corresponding minimizing points of p λ (x 1 ,...,x n ) for λ ↓ 0, see [48], [64].
Note that there may be more than one point at which the value of the global
minimum is obtained.
Example 4.1. The function: p λ (x 1 ,x 2 )=λ(x 6 1+x 6 2)+10x 2 1x 2 2−10x 2 1−10x 2 2+5 with λ =
0.5 has multiple points at which the global minimum is obtained, as shown in Figure
4.1. The global minimum value of −12.2133 is attained at 4 positions ((±1.6069, 0)
and (0, ±1.6069)). The method proposed in this chapter finds at least one point in
every connected component of the set of points where this global minimum is attained.
Figure 4.1: Example of a polynomial with multiple locations of the global optimum