[Ipopt] Ipopt Digest, Vol 114, Issue 7

[Andrew R Conn]

<<Mathematically (and I think algorithmically) these are equivalent

Not necessarily. Depends very much on the context

eg 

min_x max_i f_i(x)

is non-smooth

cf with

min_x,z z

subject to  z>= f_i(x)

for an example


Andrew R. Conn
BM T. J. Watson Research Center



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Date:   06/13/2014 12:03 PM
Subject:        Ipopt Digest, Vol 114, Issue 7
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Today's Topics:

   1. Re: question on formulation of the objective function and
      IPOPT performance (Greg Horn)


----------------------------------------------------------------------

Message: 1
Date: Fri, 13 Jun 2014 14:36:51 +0200
From: Greg Horn <gregmainland at gmail.com>
To: Damien <damien at khubla.com>
Cc: ipopt mailing list <ipopt at list.coin-or.org>
Subject: Re: [Ipopt] question on formulation of the objective function
                 and IPOPT performance
Message-ID:
 <CAAr-h4v3uuR=cHmXqgNbBSXO_v4PGE7aLNGg8qdn+S1sTFvsPg at mail.gmail.com>
Content-Type: text/plain; charset="utf-8"

Hi Damien,

Since no one has answered I'll take a shot at it, though it's not my
expertise. The case you described is:

minimize f(x) w.r.t {x}
vs
minimize y w.r.t {x,y}, subject to y == f(x)

I'm not sure if there will be any difference in this case. However, I have
had personal experience with:

minimize f(g(x)) w.r.t {x}      [prob1]
vs
minimize f(y) w.r.t. {x, y} subject to y == g(x)    [prob2]

In this case you can have significantly better convergence by "lifting"
these variables out. [prob2] will compute a better search direction than
[prob1]. This is why people use direct multiple shooting instead of direct
single shooting sometimes. There are even specialized lifting solvers 
which
can do linear algebra in the space of [prob1] and obtain the search
direction in the space of [prob2], for example:
http://num.math.uni-bayreuth.de/en/conferences/ompc_2013/program/download/friday/Diehl_ompc2013.pdf


Hope this helps,
Greg


On Wed, Jun 11, 2014 at 8:52 PM, Damien <damien at khubla.com> wrote:

> All,
>
> I'm looking at two ways to formulate the objective function in a new
> optimisation model.  The first way is what you'd call the conventional 
way
> I suppose, where you have a variety of variables contributing to the
> objective value and you calculate the gradient and return that to IPOPT.
>  The other way I'm considering is to equate the objective function to a 
new
> variable in an extra equality constraint, and have the new variable as 
the
> only variable in the objective, with a gradient of 1.0.  The new 
equality
> constraint then contributes first partial derivatives like any other
> equation or constraint.
>
> Mathematically (and I think algorithmically) these are equivalent, but I
> was wondering if anyone who's done this before has seen a performance
> difference between the two.
>
> Cheers,
>
> Damien
> _______________________________________________
> Ipopt mailing list
> Ipopt at list.coin-or.org
> http://list.coin-or.org/mailman/listinfo/ipopt
>
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