<div>min |x|</div><div><br></div><div>is equivalent to</div><div><br></div><div>min y</div><div>s.t. -y <= x <= y, y >= 0</div><div><br></div><div>and equivalent to</div><div><br></div><div><div>min y + z</div><div>
s.t. x = y - z, (y,z) >= 0</div></div><div><br></div><div>The latter two are smooth constrained problems.</div><div><br></div>Frank E. Curtis<br>P. C. Rossin Assistant Professor<br>Industrial and Systems Engineering<br>
Lehigh University<br><a href="http://coral.ie.lehigh.edu/~frankecurtis" target="_blank">http://coral.ie.lehigh.edu/~frankecurtis</a><br>
<br><br><div class="gmail_quote">On Wed, Apr 11, 2012 at 2:00 PM, Frank Kampas <span dir="ltr"><<a href="mailto:fkampas@msn.com">fkampas@msn.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
I think the technique in question is sometimes referred to as "goal programming".<br>
<br>
-----Original Message----- From: Paul van Hoven<br>
Sent: Wednesday, April 11, 2012 1:51 PM<br>
To: Peter Carbonetto<br>
Cc: <a href="mailto:ipopt@list.coin-or.org" target="_blank">ipopt@list.coin-or.org</a><br>
Subject: Re: [Ipopt] IPOpt for l1 optimization?<div class="HOEnZb"><div class="h5"><br>
<br>
Thank you for the answer Peter. Can you recommend some sources on this<br>
topic of transformation?<br>
<br>
Am 11. April 2012 18:33 schrieb Peter Carbonetto <<a href="mailto:pcarbo@uchicago.edu" target="_blank">pcarbo@uchicago.edu</a>>:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Is there an absolute value in that objective function you are minimizing? If<br>
so, then the answer is no, because the objective is non-smooth (it has<br>
undefined derivatives at zeros). But you can convert this to an equivalent<br>
smooth optimization problem with additional inequality constraints. There is<br>
quite a bit of literature on this topic.<br>
<br>
Peter Carbonetto, Ph.D.<br>
Postdoctoral Fellow<br>
Dept. of Human Genetics<br>
University of Chicago<br>
<br>
<br>
On Wed, 11 Apr 2012, Paul van Hoven wrote:<br>
<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
I've got the following problem:<br>
<br>
min_x sum_{i=1}^N | <x,c_i> |<br>
s.t. Ax < 0<br>
<br>
<x,c_i> denotes the standard scalar product between x and c_i.<br>
<br>
Is this a problem that can be solved appropriately with IPOpt?<br>
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</blockquote>
<br>
</blockquote>
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