Re: how to solve this kind of optimization problem

Hi, thanks for your reply.

I am sorry that I make a mismake in my last post. The new constrain
whould be:
sum(k(x[i],x[i+1]))<=50; i = 1, 2, 3,...99;
where k(x[i],x[i+1])=1 if x[i]~=x[i+1] or
k(x[i],x[i+1])=0 if x[i]==x[i+1].

Also the exact meaning of this is that we expect the number of changes
on x[i] is less than a constant value. For example, if
x[1]=x[2]=x[3]=...=x[99]~=x[100], then sum(k(x[i],x[i+1]))= 1 in this
case.


On 10 22 , 5 25 , spellu...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
(Peter Spellucci) wrote:
In article <1193022469.259298.26...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>, Sam <hollowsp...@xxxxxxxxx> writes:

>Hi, there.
>
>I wish I can find some help in this forum.
>
>First I have an optimization problem as follows.
>
>min f(x[1], x[2], x[3], x[4],..., x[99], x[100])
>s.t. h(x[1], x[2], x[3], x[4],..., x[99], x[100]) = 0;
> g-<g(x[1], x[2], x[3], x[4],..., x[99], x[100]) <g+;
> x-<x[i]<x+, i =1, 2, 3,..., 100;
>
>This can be solved by interior-point method.
>
>However suppose that there is a new constraint on x now.
>sum(k(x[i],x[i+1]))<=50; i = 1, 2, 3,...99;
>where k(x[i],x[i+1])=0 if x[i]~=x[i+1] and k(x[i],x[i+1])=0 if
>x[i]==x[i+1]

something is a misprint here
maybe you meant "=1" if x[i] not= x[i+1]

? no x[i] should currently change by more than 50 or the total sum?
anyway: a discontinuous constraint like your cannot be used within
nonlinear optimizeers which not only assume continuity but even
C2 - smoothness (otherwise they might terribly fail)
if you want to avoid a too large change in the current x[i]
- whatever its value might be in the given bounds - then you
could do this only by going into the code:
an trust region optimizer would allow this by setting the maximum
trust region radius -
or you could for example try
sqrt ( sum( (x[i]-x[i+1])^2 ) ) <= 50
this is also unsmooth by only on the diagonal in R^n, so its not quite
probable that this will have a bad influence

>
>The physical meaning of this new constraint is that we expect the
>change on x[i] is less than a constaint value, which is 50 in this
>example.
>
>How to deal with this new constraint? Can this new optimization
>problem still be solved by interior point method? Please do me a
>favour.
>
>Thanks in advance.
>


.

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