# [Ipopt] IPOpt for l1 optimization?

Frank E. Curtis frank.e.curtis at gmail.com
Wed Apr 11 14:29:17 EDT 2012

```min |x|

is equivalent to

min y
s.t. -y <= x <= y, y >= 0

and equivalent to

min y + z
s.t. x = y - z, (y,z) >= 0

The latter two are smooth constrained problems.

Frank E. Curtis
P. C. Rossin Assistant Professor
Industrial and Systems Engineering
Lehigh University
http://coral.ie.lehigh.edu/~frankecurtis

On Wed, Apr 11, 2012 at 2:00 PM, Frank Kampas <fkampas at msn.com> wrote:

> I think the technique in question is sometimes referred to as "goal
> programming".
>
> -----Original Message----- From: Paul van Hoven
> Sent: Wednesday, April 11, 2012 1:51 PM
> To: Peter Carbonetto
> Cc: ipopt at list.coin-or.org
> Subject: Re: [Ipopt] IPOpt for l1 optimization?
>
>
> Thank you for the answer Peter. Can you recommend some sources on this
> topic of transformation?
>
> Am 11. April 2012 18:33 schrieb Peter Carbonetto <pcarbo at uchicago.edu>:
>
>> Is there an absolute value in that objective function you are minimizing?
>> If
>> so, then the answer is no, because the objective is non-smooth (it has
>> undefined derivatives at zeros). But you can convert this to an equivalent
>> smooth optimization problem with additional inequality constraints. There
>> is
>> quite a bit of literature on this topic.
>>
>> Peter Carbonetto, Ph.D.
>> Postdoctoral Fellow
>> Dept. of Human Genetics
>> University of Chicago
>>
>>
>> On Wed, 11 Apr 2012, Paul van Hoven wrote:
>>
>>  I've got the following problem:
>>>
>>> min_x sum_{i=1}^N | <x,c_i> |
>>> s.t. Ax < 0
>>>
>>> <x,c_i> denotes the standard scalar product between x and c_i.
>>>
>>> Is this a problem that can be solved appropriately with IPOpt?
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