Thread Subject: Minimization of the sum of integrals with unknown bounds

Hi all,

I have to solve a pretty messed up problem such:


min_y sum_{h=1}^H int(-inf,y(h)) fun(y)


where y is the H-dimensional vector of the elements y(h).

Does anyone know how to solve such a problem?

Thank you in advance!

Ita

On 8 Jul., 13:06, "Ita Atz" <ita....@gmail.com> wrote:
> Hi all,
>
> I have to solve a pretty messed up problem such:
>
> min_y sum_{h=1}^H int(-inf,y(h)) fun(y)
>
> where y is the H-dimensional vector of the elements y(h).
>
> Does anyone know how to solve such a problem?
>
> Thank you in advance!
>
> Ita

You mean
min_y sum_{h=1}^{H} int(-inf,y(h)) fun(y_1,...,t,...y_H) dt
where the t is at the h-th position ?

Best wishes
Torsten.

Torsten <Torsten.Hennig@umsicht.fraunhofer.de> wrote in message <e4a50d37-128d-445a-b943-b627e6fa6d00@t5g2000yqj.googlegroups.com>...
> On 8 Jul., 13:06, "Ita Atz" <ita....@gmail.com> wrote:
> > Hi all,
> >
> > I have to solve a pretty messed up problem such:
> >
> > min_y sum_{h=1}^H int(-inf,y(h)) fun(y)
> >
> > where y is the H-dimensional vector of the elements y(h).
> >
> > Does anyone know how to solve such a problem?
> >
> > Thank you in advance!
> >
> > Ita
>
> You mean
> min_y sum_{h=1}^{H} int(-inf,y(h)) fun(y_1,...,t,...y_H) dt
> where the t is at the h-th position ?
>
> Best wishes
> Torsten.

Hi, thanks for replying!

I had to write it more in detail:

min_y sum_{h=1}^H int(-inf,y(h)) fun(x,y) dx

my unknown vector variable is y which appears in the integrand, while each one of its H elements appears in the upper bound of each one of the H integral in the sum. I have to find the vector y that minimizes the sum of those integrals.

I know, it's sick...

Ita

On 8 Jul., 13:36, "Ita Atz" <ita....@gmail.com> wrote:
> Torsten <Torsten.Hen...@umsicht.fraunhofer.de> wrote in message <e4a50d37-128d-445a-b943-b627e6fa6...@t5g2000yqj.googlegroups.com>...
> > On 8 Jul., 13:06, "Ita Atz" <ita....@gmail.com> wrote:
> > > Hi all,
>
> > > I have to solve a pretty messed up problem such:
>
> > > min_y sum_{h=1}^H int(-inf,y(h)) fun(y)
>
> > > where y is the H-dimensional vector of the elements y(h).
>
> > > Does anyone know how to solve such a problem?
>
> > > Thank you in advance!
>
> > > Ita
>
> > You mean
> > min_y sum_{h=1}^{H} int(-inf,y(h)) fun(y_1,...,t,...y_H) dt
> > where the t is at the h-th position ?
>
> > Best wishes
> > Torsten.
>
> Hi, thanks for replying!
>
> I had to write it more in detail:
>
> min_y sum_{h=1}^H int(-inf,y(h)) fun(x,y) dx
>
> my unknown vector variable is y which appears in the integrand, while each one of its H elements appears in the upper bound of each one of the H integral in the sum. I have to find the vector y that minimizes the sum of those integrals.
>
> I know, it's sick...
>
> Ita- Zitierten Text ausblenden -
>
> - Zitierten Text anzeigen -


And x is one-dimensional ?

Best wishes
Torsten.

Torsten <Torsten.Hennig@umsicht.fraunhofer.de> wrote in message <2d12d43f-5b32-4551-95bc-cfcc3fa8f442@gv8g2000vbb.googlegroups.com>...
> On 8 Jul., 13:36, "Ita Atz" <ita....@gmail.com> wrote:
> > Torsten <Torsten.Hen...@umsicht.fraunhofer.de> wrote in message <e4a50d37-128d-445a-b943-b627e6fa6...@t5g2000yqj.googlegroups.com>...
> > > On 8 Jul., 13:06, "Ita Atz" <ita....@gmail.com> wrote:
> > > > Hi all,
> >
> > > > I have to solve a pretty messed up problem such:
> >
> > > > min_y sum_{h=1}^H int(-inf,y(h)) fun(y)
> >
> > > > where y is the H-dimensional vector of the elements y(h).
> >
> > > > Does anyone know how to solve such a problem?
> >
> > > > Thank you in advance!
> >
> > > > Ita
> >
> > > You mean
> > > min_y sum_{h=1}^{H} int(-inf,y(h)) fun(y_1,...,t,...y_H) dt
> > > where the t is at the h-th position ?
> >
> > > Best wishes
> > > Torsten.
> >
> > Hi, thanks for replying!
> >
> > I had to write it more in detail:
> >
> > min_y sum_{h=1}^H int(-inf,y(h)) fun(x,y) dx
> >
> > my unknown vector variable is y which appears in the integrand, while each one of its H elements appears in the upper bound of each one of the H integral in the sum. I have to find the vector y that minimizes the sum of those integrals.
> >
> > I know, it's sick...
> >
> > Ita- Zitierten Text ausblenden -
> >
> > - Zitierten Text anzeigen -
>
>
> And x is one-dimensional ?
>
> Best wishes
> Torsten.

No, x is H-dimensional, in fact the function to be minimized could be written as


sum_{h=1}^H ( int(-inf,y(h)) fun(x(h),y) dx(h) )


Ita

On 8 Jul., 14:00, "Ita Atz" <ita....@gmail.com> wrote:
> Torsten <Torsten.Hen...@umsicht.fraunhofer.de> wrote in message <2d12d43f-5b32-4551-95bc-cfcc3fa8f...@gv8g2000vbb.googlegroups.com>...
> > On 8 Jul., 13:36, "Ita Atz" <ita....@gmail.com> wrote:
> > > Torsten <Torsten.Hen...@umsicht.fraunhofer.de> wrote in message <e4a50d37-128d-445a-b943-b627e6fa6...@t5g2000yqj.googlegroups.com>...
> > > > On 8 Jul., 13:06, "Ita Atz" <ita....@gmail.com> wrote:
> > > > > Hi all,
>
> > > > > I have to solve a pretty messed up problem such:
>
> > > > > min_y sum_{h=1}^H int(-inf,y(h)) fun(y)
>
> > > > > where y is the H-dimensional vector of the elements y(h).
>
> > > > > Does anyone know how to solve such a problem?
>
> > > > > Thank you in advance!
>
> > > > > Ita
>
> > > > You mean
> > > > min_y sum_{h=1}^{H} int(-inf,y(h)) fun(y_1,...,t,...y_H) dt
> > > > where the t is at the h-th position ?
>
> > > > Best wishes
> > > > Torsten.
>
> > > Hi, thanks for replying!
>
> > > I had to write it more in detail:
>
> > > min_y sum_{h=1}^H int(-inf,y(h)) fun(x,y) dx
>
> > > my unknown vector variable is y which appears in the integrand, while each one of its H elements appears in the upper bound of each one of the H integral in the sum. I have to find the vector y that minimizes the sum of those integrals.
>
> > > I know, it's sick...
>
> > > Ita- Zitierten Text ausblenden -
>
> > > - Zitierten Text anzeigen -
>
> > And x is one-dimensional ?
>
> > Best wishes
> > Torsten.
>
> No, x is H-dimensional, in fact the function to be minimized could be written as
>
> sum_{h=1}^H ( int(-inf,y(h)) fun(x(h),y) dx(h) )
>
> Ita- Zitierten Text ausblenden -
>
> - Zitierten Text anzeigen -


Still not clear to me. An example would be useful, I think.

Best wishes
Torsten.

Torsten <Torsten.Hennig@umsicht.fraunhofer.de> wrote in message <a3a75740-7ae9-4b13-9413-2144d36a8c5a@ct4g2000vbb.googlegroups.com>...
> On 8 Jul., 14:00, "Ita Atz" <ita....@gmail.com> wrote:
> > Torsten <Torsten.Hen...@umsicht.fraunhofer.de> wrote in message <2d12d43f-5b32-4551-95bc-cfcc3fa8f...@gv8g2000vbb.googlegroups.com>...
> > > On 8 Jul., 13:36, "Ita Atz" <ita....@gmail.com> wrote:
> > > > Torsten <Torsten.Hen...@umsicht.fraunhofer.de> wrote in message <e4a50d37-128d-445a-b943-b627e6fa6...@t5g2000yqj.googlegroups.com>...
> > > > > On 8 Jul., 13:06, "Ita Atz" <ita....@gmail.com> wrote:
> > > > > > Hi all,
> >
> > > > > > I have to solve a pretty messed up problem such:
> >
> > > > > > min_y sum_{h=1}^H int(-inf,y(h)) fun(y)
> >
> > > > > > where y is the H-dimensional vector of the elements y(h).
> >
> > > > > > Does anyone know how to solve such a problem?
> >
> > > > > > Thank you in advance!
> >
> > > > > > Ita
> >
> > > > > You mean
> > > > > min_y sum_{h=1}^{H} int(-inf,y(h)) fun(y_1,...,t,...y_H) dt
> > > > > where the t is at the h-th position ?
> >
> > > > > Best wishes
> > > > > Torsten.
> >
> > > > Hi, thanks for replying!
> >
> > > > I had to write it more in detail:
> >
> > > > min_y sum_{h=1}^H int(-inf,y(h)) fun(x,y) dx
> >
> > > > my unknown vector variable is y which appears in the integrand, while each one of its H elements appears in the upper bound of each one of the H integral in the sum. I have to find the vector y that minimizes the sum of those integrals.
> >
> > > > I know, it's sick...
> >
> > > > Ita- Zitierten Text ausblenden -
> >
> > > > - Zitierten Text anzeigen -
> >
> > > And x is one-dimensional ?
> >
> > > Best wishes
> > > Torsten.
> >
> > No, x is H-dimensional, in fact the function to be minimized could be written as
> >
> > sum_{h=1}^H ( int(-inf,y(h)) fun(x(h),y) dx(h) )
> >
> > Ita- Zitierten Text ausblenden -
> >
> > - Zitierten Text anzeigen -
>
>
> Still not clear to me. An example would be useful, I think.
>
> Best wishes
> Torsten.

I'll try to describe my specific problem.

I have a H-dimensional vector x, whose elements x(h) are random variables with known distributions pdf(x(h)). So I have to find the H-dimensional vector y that minimizes


sum_{h=1}^H [ int(-inf,y(h)) pdf(x(h)) fun(x(h),y) dx(h) ]


where the function fun is known and depends both on the single element of x and the whole vector y, which is the one I have to calculate in the minimization.

I hope it was clear. Thanks for your patience!

Ita

"Ita Atz" <ita.atz@gmail.com> wrote in message
news:iv6up8$rp4$1@newscl01ah.mathworks.com...
> Torsten <Torsten.Hennig@umsicht.fraunhofer.de> wrote in message
> <a3a75740-7ae9-4b13-9413-2144d36a8c5a@ct4g2000vbb.googlegroups.com>...

*snip*

> I'll try to describe my specific problem.
>
> I have a H-dimensional vector x, whose elements x(h) are random variables
> with known distributions pdf(x(h)). So I have to find the H-dimensional
> vector y that minimizes
>
>
> sum_{h=1}^H [ int(-inf,y(h)) pdf(x(h)) fun(x(h),y) dx(h) ]
>
>
> where the function fun is known and depends both on the single element of
> x and the whole vector y, which is the one I have to calculate in the
> minimization.
>
> I hope it was clear. Thanks for your patience!

Not really. Show a SPECIFIC example where H = 3 (for demonstration purposes;
your real value of H may be much larger, but let's stick with a value that
fits in a newsgroup posting.) Describe exactly how you generate x and give a
simple example of a fun function, then show the specific expression that you
want to minimize.

--
Steve Lord
slord@mathworks.com
To contact Technical Support use the Contact Us link on
http://www.mathworks.com

"Ita Atz" wrote in message <iv6ob2$b6t$1@newscl01ah.mathworks.com>...
> Hi all,
> I have to solve a pretty messed up problem such:
>
> min_y sum_{h=1}^H int(-inf,y(h)) fun(y)
>
> where y is the H-dimensional vector of the elements y(h).
> Does anyone know how to solve such a problem?
> Thank you in advance!
> Ita
- - - - - - - - - - -
  You haven't made your problem at all clear to me either, Ita. I think you better keep on trying to improve your description. You can do better than you have up to this point. However, the following fact may (or may not) be of use to you.

  The minimum of a single integral with respect to variation of its upper limit of integration, assuming nothing else varies but the limit, must occur either at a point where its integrand is zero or else at an extreme value in the range allowed for that limit. You can show this by taking the derivative of the integral which would be its integrand evaluated at the upper limit.

  Of course having a zero integrand does not guarantee a minimum. It might be a maximum. It depends on whether the integrand is descending or ascending at that point.

  In your sum of integrals if you are able to vary each upper limit independently of the others, then the minimum must occur in such circumstances in each integral independently of the others.

Roger Stafford