Speaker: Bernard Hanzon
Joined work with Ivo Bleylevens and Ralf L.M. Peeters
Title: A multidimensional systems approach to algebraic optimization
Abstract:
With any system of multivariate polynomial equations we can associate a
system of multidimensional difference equations by interpreting
the variables in the polynomial equations as shift operators
working on a multidimensional time series. If the solution set of
the system of multivariate polynomial equations is finite,
including possible complex solutions, then the associated system
of multidimensional difference equations has a finite dimensional
solution set and one can describe the solution set as the outcomes
of a finite dimensional state space system, a so-called state
space realization of the system of multidimensional difference
equations. Scalar solutions of the original system of polynomial
equations coincide with the multi-poles of such a state-space
realization, provided the state-space realization is minimal, i.e.
has smallest possible state-space dimension. Each multi-pole is a
multi-eigenvalue of the tuple of matrices representing the various
shift operators. A multi-eigenvalue corresponds to a well-defined
eigenspace. Here a special class of polynomials is considered for
which the global minimization problem is solved by associating a
multidimensional system to the first order conditions for
minimization of the polynomial and solving for the first order
conditions using the above method. The method leads to large
eigenvalue problems. One way to solve them is by Jacobi-Davidson
methods. For such methods one does not need to calculate the
matrix that is involved explicitly, one only needs to specify the
action of the linear operator on any given vector. Using the
multidimensional system structure the action of the linear
operator can be obtained by solving for certain elements of the
multidimensional sequence that forms the solution of the system of
difference equations given a sufficient number of initial values
of the sequence.
To illustrate the theory and to show how the method works in practice,
some examples will be presented.
One such example concerns an optimal $H_2$ model order reduction problem
in systems theory.
Some remarks will be made about possibilities for further improvement of
the various software modules for this kind of application.
References:
B. Hanzon and D. Jibetean,
Global minimization of a multivariate polynomial using matrix
methods, Journal of Global Optimization, 27, 1--23,
September 2003
B. Hanzon and J.M. Maciejowski,
Constructive algebra methods for the $L_2$ problem for stable linear
systems, Automatica, Vol. 32, No. 12, 1645--1657, 1996.
B. Hanzon, J.M. Maciejowski, C.T. Chou,
Model reduction in H2 using matrix solutions of polynomial
equations,
Cambridge University Department of Engineering Technical Reports
CUED/F-INFENG/TR.314, Cambridge UK, March 1998
I. Bleylevens, B. Hanzon, R.L.M. Peeters, A multidimensional systems
approach to polynomial optimization,
forthcoming.