Re: optimization problem -- any pointers?


In article <e253a592-b4b5-4e18-9807-05074e94767f@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Hyped <nortonsm@xxxxxxxxx> writes:
On Feb 26, 10:29=A0pm, Hyped <norto...@xxxxxxxxx> wrote:
Hi,
I have a problem that I'm hoping to get some direction on.
It's an optimization problem but, other than that, I'm having trouble
determing which textbooks to pick up to get started.
The idea is the following. =A0I have a given spectra designated S1,
which is composed of n points. =A0These n points are intensity values,
all positive.
My aim is to determine another spectra, called S2 which is "most
orthogonal" to S1. =A0That is,
maximize <S1 | S2 >
"most orthogonal" means <S1 | S2 > =0
so what is to maximize or minimize here??



As I was saying: Maximize <S1 | S2> given the following constraints:
1) the total energy under S2 matches S1 -- i.e. the L2-norm of S1 is
the same as S2
sum(S1(i)^2) (given) = sum S2(i)^2

2) all intensities of the n-points of S2 > 0

S2(i) >=eps>0 for all i (there is no possibility for a ">" )
3) max bandwith of S2 =3D max bandwidth of S1
#3 is a constraint on spectral resolution. Since the spectras are
captured by a spectrometer, one can't have a higher bandwidth than
allowed by the slit of the spectrometer. In other words, I can't have
peaks that are narrower than some limit. Constraint #3 is difficult
for me to understand how to invoke in say, Linear Programming methods
of optimization. #3 is, in a sense, a constraint in the fourier space
while the rest of the problem happens in wavelength (or call it "time"
or non-fourier space).

Anyways, I have little idea how to attack this problem or where to
find help in this optimization problem. It appears I should focus on
Linear Programming literature, but a quick perview has resulted in my
not finding anything that has fourier constraints.

Any pointers, references, appreciated.
Thanks and apologize the posting got split up.
-Hyped

A= discrete Fourier(S2) (a linear transformation)

A(i) = 0 if i>= n n given and << N , N= dimension of S1 ??
is it this what you mean?

hth
peter

.

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