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Chapter 4
Global optimization of multivariate
polynomials
Finding the global minimum of a real-valued multivariate polynomial is a problem
which has several useful applications in systems and control theory as well as in many
other quantitative sciences including statistics, mathematical finance, economics, systems
biology, etc. Multivariate global polynomial optimization is a hard problem because
of the non-convexity of the problem and the existence of local optima. In this
chapter we focus on computing the global optimum of a special class of polynomials.
The class of polynomials discussed in this chapter is the class of so-called Minkowski
dominated polynomials in several variables. When working with a polynomial p λ (x 1 ,
...,x n ) from this class, the problem of finding its global minimum can be reformulated
as an eigenvalue problem by applying the Stetter-Möller matrix method introduced in
the previous chapter. This is possible because the system of first-order conditions of
such a polynomial is always in Gröbner basis form with a special structure, showing
that the number of solutions is finite. Applying the Stetter-Möller matrix method
yields a set of real non-symmetric large and sparse commuting matrices A T x 1
,...,A T x n
.
Using the same approach, it is also possible to construct a matrix A T p λ (x which
1,...,x n)
has some useful properties: it turns out that the smallest real eigenvalue of the matrix
A T p λ x 1,...,x n)
is of interest for computing the global optimum of the polynomial
p λ (x 1 ,...,x n ).
Various extensions and generalizations of these techniques exist including the optimization
of rational or logarithmic functions [63] and the use of systems that are
not in Gröbner basis form [12]. However, these generalizations do not satisfy all the
properties described below and therefore require adjustments to the techniques used
here.
Numerical experiments using the approach described in this chapter are discussed
in Chapter 7.
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