<tt><font size=2><<Mathematically (and I think algorithmically)
these are equivalent</font></tt>
<br>
<br><font size=2 face="sans-serif">Not necessarily. Depends very much on
the context</font>
<br>
<br><font size=2 face="sans-serif">eg </font>
<br>
<br><font size=2 face="sans-serif">min_x max_i f_i(x)</font>
<br>
<br><font size=2 face="sans-serif">is non-smooth</font>
<br>
<br><font size=2 face="sans-serif">cf with</font>
<br>
<br><font size=2 face="sans-serif">min_x,z z</font>
<br>
<br><font size=2 face="sans-serif">subject to z>= f_i(x)</font>
<br>
<br><font size=2 face="sans-serif">for an example</font>
<br>
<br>
<br><font size=2 color=blue face="Arial">Andrew R. Conn</font>
<br><font size=2 color=blue face="Arial">BM T. J. Watson Research Center</font>
<br>
<br>
<br>
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</font>
<br><font size=1 color=#5f5f5f face="sans-serif">Date:
</font><font size=1 face="sans-serif">06/13/2014 12:03 PM</font>
<br><font size=1 color=#5f5f5f face="sans-serif">Subject:
</font><font size=1 face="sans-serif">Ipopt Digest,
Vol 114, Issue 7</font>
<br><font size=1 color=#5f5f5f face="sans-serif">Sent by:
</font><font size=1 face="sans-serif">ipopt-bounces@list.coin-or.org</font>
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Today's Topics:<br>
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1. Re: question on formulation of the objective function and<br>
IPOPT performance (Greg Horn)<br>
<br>
<br>
----------------------------------------------------------------------<br>
<br>
Message: 1<br>
Date: Fri, 13 Jun 2014 14:36:51 +0200<br>
From: Greg Horn <gregmainland@gmail.com><br>
To: Damien <damien@khubla.com><br>
Cc: ipopt mailing list <ipopt@list.coin-or.org><br>
Subject: Re: [Ipopt] question on formulation of the objective function<br>
and IPOPT performance<br>
Message-ID:<br>
<CAAr-h4v3uuR=cHmXqgNbBSXO_v4PGE7aLNGg8qdn+S1sTFvsPg@mail.gmail.com><br>
Content-Type: text/plain; charset="utf-8"<br>
<br>
Hi Damien,<br>
<br>
Since no one has answered I'll take a shot at it, though it's not my<br>
expertise. The case you described is:<br>
<br>
minimize f(x) w.r.t {x}<br>
vs<br>
minimize y w.r.t {x,y}, subject to y == f(x)<br>
<br>
I'm not sure if there will be any difference in this case. However, I have<br>
had personal experience with:<br>
<br>
minimize f(g(x)) w.r.t {x} [prob1]<br>
vs<br>
minimize f(y) w.r.t. {x, y} subject to y == g(x) [prob2]<br>
<br>
In this case you can have significantly better convergence by "lifting"<br>
these variables out. [prob2] will compute a better search direction than<br>
[prob1]. This is why people use direct multiple shooting instead of direct<br>
single shooting sometimes. There are even specialized lifting solvers which<br>
can do linear algebra in the space of [prob1] and obtain the search<br>
direction in the space of [prob2], for example:<br>
</font></tt><a href="http://num.math.uni-bayreuth.de/en/conferences/ompc_2013/program/download/friday/Diehl_ompc2013.pdf"><tt><font size=2>http://num.math.uni-bayreuth.de/en/conferences/ompc_2013/program/download/friday/Diehl_ompc2013.pdf</font></tt></a><tt><font size=2><br>
<br>
Hope this helps,<br>
Greg<br>
<br>
<br>
On Wed, Jun 11, 2014 at 8:52 PM, Damien <damien@khubla.com> wrote:<br>
<br>
> All,<br>
><br>
> I'm looking at two ways to formulate the objective function in a new<br>
> optimisation model. The first way is what you'd call the conventional
way<br>
> I suppose, where you have a variety of variables contributing to the<br>
> objective value and you calculate the gradient and return that to
IPOPT.<br>
> The other way I'm considering is to equate the objective function
to a new<br>
> variable in an extra equality constraint, and have the new variable
as the<br>
> only variable in the objective, with a gradient of 1.0. The
new equality<br>
> constraint then contributes first partial derivatives like any other<br>
> equation or constraint.<br>
><br>
> Mathematically (and I think algorithmically) these are equivalent,
but I<br>
> was wondering if anyone who's done this before has seen a performance<br>
> difference between the two.<br>
><br>
> Cheers,<br>
><br>
> Damien<br>
> _______________________________________________<br>
> Ipopt mailing list<br>
> Ipopt@list.coin-or.org<br>
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