Probl. Lin.-Algeb./Optim.

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2 total messages Started by Rainer Bachl Fri, 20 Nov 1992 03:51

#910

Author: Rainer Bachl
Date: Fri, 20 Nov 1992 03:51

30 lines
1571 bytes

After some efforts and pages of manipulations of an engineering problem
in signal processing I have got a nice mathematical formulation for it:

Find a real vector d constrained to a convex subspace subject to

             min | A * P |
                          2

where P is a real orthogonal projector on the subspace          
of span{ diag(d) * H }, i.e.

                                 T        2  -1    T
             P = diag(d) * H * {H *diag(d) H}   * H  * diag(d)

and A,H are both known real matrices. However, H is m*n with m>n and
satisfies
              T
             H  * H = identity(n).

I think there exists no closed form solution and therefore I am looking
for an optimization procedure locally (if not globally) minimizing the
above cost function. Iterative schemes, similar to that of Steiglitz/
McBride or the IQML algorithm, whereby d(i+1) is calculated by using
a fixed d(i) in the matrix inverse, do not converge to a local minimum
close to the initial value of d. Methods involving first and second
order derivatives might be computationally too expensive for this
application.

I am not an expert in this area and any suggestions are welcome.

Re: Probl. Lin.-Algeb./Optim.

#942

Author: Rainer Bachl
Date: Sat, 21 Nov 1992 10:28

37 lines
1938 bytes

engp2116@nuscc.nus.sg (Rainer Bachl) writes:
: 
: After some efforts and pages of manipulations of an engineering problem
: in signal processing I have got a nice mathematical formulation for it:
: 
: Find a real vector d constrained to a convex subspace subject to
: 
:              min | A * P |
:                           2
: 
: where P is a real orthogonal projector on the subspace          
: of span{ diag(d) * H }, i.e.
: 
:                                  T        2  -1    T
:              P = diag(d) * H * {H *diag(d) H}   * H  * diag(d)
: 
: and A,H are both known real matrices. However, H is m*n with m>n and
: satisfies
:               T
:              H  * H = identity(n).
: 
: I think there exists no closed form solution and therefore I am looking
: for an optimization procedure locally (if not globally) minimizing the
: above cost function. Iterative schemes, similar to that of Steiglitz/
: McBride or the IQML algorithm, whereby d(i+1) is calculated by using
: a fixed d(i) in the matrix inverse, do not converge to a local minimum
: close to the initial value of d. Methods involving first and second
: order derivatives might be computationally too expensive for this
: application.
: 
: I am not an expert in this area and any suggestions are welcome.

Sorry, forgot to mention an arbitrary norm constraint for the desired
solution, e.g. | d |  = 1, to avoid the trivial case d = 0.
                    2
-Rainer Bachl
 National University of Singapore

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