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<div class="moz-cite-prefix">Sorry. By the "kernel", I mean the
kernel of the Jacobian of f(x):<br>
<br>
All u in R^n such that df/dx * u = 0<br>
<br>
These u form the tangent space of the constraint "surface".<br>
You could, for example, use SVD to find an orthonormal basis of
the kernel.<br>
<br>
Cheers,<br>
Norm<br>
<br>
<br>
On 04/09/2013 01:11 AM, Ingrid Hallen wrote:<br>
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<div dir="ltr">Ok, I think I kind of understand the idea. However,
doesn't it require<br>
that one solves equations of the form f(x) = 0, where f define<br>
some of the constraints, for each iterate x to identify the
kernel? <br>
It seems related to the idea of eliminating equality constraints
(as<br>
in e.g. p. 132 in Convex Optimization by Boyd and Vandenberge),<br>
which, as I did not mention :), I don't want to do.<br>
<br>
<br>
Kind regards,<br>
<br>
Ingrid<br>
<br>
<br>
<br>
<br>
<br>
<br>
<div>
<hr id="stopSpelling">Date: Mon, 8 Apr 2013 13:14:39 -0700<br>
From: <a class="moz-txt-link-abbreviated" href="mailto:normvcr@telus.net">normvcr@telus.net</a><br>
To: <a class="moz-txt-link-abbreviated" href="mailto:ingridhallen@hotmail.com">ingridhallen@hotmail.com</a><br>
CC: <a class="moz-txt-link-abbreviated" href="mailto:adol-c@list.coin-or.org">adol-c@list.coin-or.org</a><br>
Subject: Re: [ADOL-C] Evaluating Hessian of Lagrangian<br>
<br>
<div class="ecxmoz-cite-prefix">For the second point,
specially since there are almost as many<br>
constraints as variables, it may be worthwhile to take the
Hessian<br>
with respect to only the directions in the kernel of the
constraints.<br>
For example, if you are working in 20-dimensional Euclidean
space,<br>
with 18 constraints, that leaves an essentially
2-dimensional optimization<br>
problem, and you only really need a 2x2 Hessian, not a 20x20
Hessian.<br>
You can choose two orthogonal directions in the kernel of
the constraints,<br>
and construct the 2x2 Hessian of Lagrangian wrt them,
viewing this either<br>
as directional derivatives or change of coordinates.<br>
<br>
I don't know what amount of SW changes this would need, but
it<br>
seems it would be generally useful.<br>
<br>
Cheers,<br>
Norm<br>
<br>
<br>
On 04/08/2013 06:38 AM, Ingrid Hallen wrote:<br>
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<div dir="ltr">Thanks for the suggestions!<br>
<br>
Regarding the first, I think it might not be so efficient
for my problem, having<br>
almost as many constraints as variables. But I might give
it a try. Perhaps<br>
one can modify the code for sparse_hess in some clever way
...<br>
<br>
Regarding the second I am unfortunately clueless as to how
one could<br>
exploit that fact.<br>
<br>
Kind regards,<br>
<br>
Ingrid<br>
<br>
<br>
<div>
<hr id="ecxstopSpelling">Date: Sun, 7 Apr 2013 15:31:50
-0700<br>
From: <a moz-do-not-send="true"
class="ecxmoz-txt-link-abbreviated"
href="mailto:normvcr@telus.net">normvcr@telus.net</a><br>
To: <a moz-do-not-send="true"
class="ecxmoz-txt-link-abbreviated"
href="mailto:adol-c@list.coin-or.org">adol-c@list.coin-or.org</a><br>
Subject: Re: [ADOL-C] Evaluating Hessian of Lagrangian<br>
<br>
<div class="ecxmoz-cite-prefix">Perhaps you can trace L
as a function of both x and lambda.<br>
Then when you need the hessian,<br>
calculate the hessian only with respect to x.<br>
Not sure if this is better/worse than what you
suggested.<br>
<br>
When using the Hessian of the Lagrangian, you will
eventually be restricting<br>
attention to the kernel of your constraints. Do you
know if this fact<br>
would offer a simplification to how the hessian could
be computed?<br>
<br>
Norm<br>
<br>
<br>
On 04/05/2013 04:02 AM, Ingrid Hallen wrote:<br>
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<div dir="ltr">Hi,<br>
<br>
I'm doing non-linear optimization with IPOPT. For
this, I'm using ADOL-C<br>
to compute the Hessian of the Lagrangian<br>
<br>
L(x,lambda) = f(x) + sum_{i}lambda_{i}h_{i}(x),<br>
<br>
where x are the variables, lambda the Lagrange
multipliers and <br>
f(x) and h_{i}(x) objective and constraint
functions.<br>
<br>
What I'm doing in my code is the following (omitting
details):<br>
<br>
// **********************<br>
<br>
// Trace Lagrangian function<br>
trace_on(tag);<br>
<br>
for(i=0;i<n;i++) {<br>
xad[i] <<= x[i];<br>
}<br>
<br>
Lagrangian(xad, lambda);<br>
<br>
Lad >>=L;<br>
<br>
trace_off();<br>
<br>
// Evaluate Hessian of the Lagrangian<br>
repeat = 0;<br>
sparse_hess(tag,n,repeat,x,&nnz,&rind,&cind,&values,&options)<br>
<br>
// ***********************<br>
<br>
This works fine, but is not so efficient. One reason
is that, since lambda changes, <br>
the Lagrangian function has to be retaped every time
the Hessian is needed and so it <br>
appears that I cannot set repeat = 1 when calling
sparse_hess.<br>
<br>
One way to circumvent this problem could perhaps be
to trace the objective<br>
and constraint functions individually and then
construct the Hessian of<br>
the Lagrangian using multiple calls to sparse_hess,
but is there a<br>
more convenient way to do it? <br>
<br>
Sincerely,<br>
<br>
Ingrid<br>
<br>
</div>
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