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7.1. COMPUTING THE MINIMUM OF A POLYNOMIAL OF ORDER 8 111
required operator actions/matrix-vector products (MVs), and the running time. For
these computations the default parameters of the various methods are used.
Table 7.2: Minimal eigenvalue of the operator A p1
(one eigenvalue calculated)
Method Eigenvalue Error #MVs Time (s)
Eigs 4.09516474405595 3.03 × 10 −10 3040 173
JDQR 4.09516474449668 1.38 × 10 −10 312 21
JDQZ 4.09516474427060 8.86 × 10 −11 302 21
Eigs with balancing 4.09516468968153 5.47 × 10 −8 2752 162
JDQR with balancing 4.09516473924322 5.12 × 10 −9 421 28
JDQZ with balancing 4.09516473593991 8.42 × 10 −9 389 28
The global minimum computed by the JDQZ method produces the critical value
that is the closest to the ‘true’ critical value computed by the local search method.
For this setting, the nD-systems approach uses the fewest operator actions. Using
the corresponding eigenvector, the coordinates of the stationary point corresponding
to this global minimum are computed as:
x 1 =+0.876539213107485
x 2 = −0.903966282291641
x 3 =+0.862027936168838
x 4 = −0.835187476763094
(7.5)
For the problem of finding the global minimum of a polynomial there are several
other specialized software packages available, which employ different approaches. To
put the performance of the nD-systems approach into perspective, we will briefly
discuss the outcomes of the computation of the global minimum of polynomial (7.1)
when using the software packages SOSTOOLS, GloptiPoly and SYNAPS.
SOSTOOLS is a Matlab toolbox for formulating and solving sum of squares (SOS)
problems (see [86], [90]). This toolbox uses the Matlab solver SeDuMi (see [96]) to
solve the involved semi-definite programs (SDP). To compute the global minimum,
SOSTOOLS searches for the largest possible γ for which p 1 (x 1 ,x 2 ,x 3 ,x 4 ) −γ is still a
sum of squares. This γ may be the global minimum p ⋆ we are looking for, depending
on whether the polynomial p 1 (x 1 ,x 2 ,x 3 ,x 4 ) − p ⋆ can be written as a sum of squares
of polynomials (see [87]). Note that the nD-systems approach does not suffer from
such a limitation.
GloptiPoly (see [54]) solves a multivariable polynomial optimization problem by
building and solving convex linear matrix inequality (LMI) relaxations of the problem
using the solver SeDuMi. It produces a series of lower bounds which converge to the
global optimum we are looking for. The theory of moments and positive polynomials
is used in the implementation of this software (see [74] and [75]).
SYNAPS [91] is a C++ environment for symbolic and numeric computations. It
provides a routine to search for the real solutions of a polynomial system of equations
like system (7.2) within a given domain (based on a subdivision method). All the