# [Ipopt] Non-convex Objective that is Convex on a (Convex) Feasible Region

Victor Wu vwwu at umich.edu
Tue Sep 13 13:10:01 EDT 2016

```I will be using a limited memory approximation.  Also, it may be difficult
to find a feasible starting solution.

It seems that a convex extension of the objective outside the feasible
region would be a safer way to guarantee global optimality?

On Mon, Sep 12, 2016 at 1:24 PM, Ipopt User <ipoptuser at gmail.com> wrote:

> I'm assuming you use the full Hessian. For a nonconvex objective, the
> Hessian of the Lagrangian may not be positive semidefinite, and some
> multiple of the identity matrix is added to it which may hinder
> convergence. If it converges to a point in the interior of the feasible
> region, you obtain a feasible primal-dual solution which guarantees global
> optimality. If it converges to a point on the boundary, that point does not
> have to be optimal; e.g., take f(x) = x^2 for x<0, f(x) = (x-2)^2-4 for
> x>=0 as the objective, and x>=0 as the feasible region; it may converge
> from the negatives to 0 (which seems like a KKT point if you approach it
> from the left since the derivative seems to become 0).
>
> On Mon, Sep 12, 2016 at 10:58 AM, Victor Wu <vwwu at umich.edu> wrote:
>
>> Hi Stefan,
>>
>> an infeasible solution, is it possible that I get trapped in a locally
>> optimal solution *just* outside the feasible region (assuming such
>> solutions exist)?
>>
>> Victor
>>
>> On Mon, Sep 12, 2016 at 10:47 AM, Stefan Vigerske <
>> stefan at math.hu-berlin.de> wrote:
>>
>>> Hi,
>>>
>>> if you provide a feasible starting point, I would imagine that
>>> nonconvexity of the objective function outside of the feasible area doesn't
>>> matter.
>>>
>>> Stefan
>>>
>>>
>>>
>>> On 09/12/2016 04:36 PM, Victor Wu wrote:
>>>
>>>> Hello,
>>>>
>>>> I have a nonlinear problem where the objective function is non-convex in
>>>> general, but is convex over the (convex) feasible region of interest
>>>> (thus,
>>>> theoretically I have a convex problem).  Do you have any
>>>> recommendations on
>>>> how I should represent the objective (i.e., write the objective in its
>>>> original non-convex form and enforce my feasible region or do something
>>>> else) in IpOpt?  My concern is whether IpOpt would reach the
>>>> *theoretically* global minimizer and if it depends on my implementation.
>>>>
>>>> Thank you,
>>>> Victor Wu
>>>>
>>>>
>>>>
>>>> _______________________________________________
>>>> Ipopt mailing list
>>>> Ipopt at list.coin-or.org
>>>> https://urldefense.proofpoint.com/v2/url?u=http-3A__list.coi
>>>> n-2Dor.org_mailman_listinfo_ipopt&d=CwICAg&c=Ngd-ta5yRYsqeUs
>>>> EDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=BRcuJnQr5NAzU29t80hk2rsLc4vrl
>>>> RySBDabuq0O1ZI&m=DHz4_ayi1_efxmdM578VdOrLBxYZHfluKOHPPd0BK5E
>>>> &s=uQ23hHdajrGJ4gLHizIm0kTYkgu1tfZ1W22iZ84mn9o&e=
>>>>
>>>>
>>>
>>> --
>>> http://www.gams.com/~stefan
>>> <https://urldefense.proofpoint.com/v2/url?u=http-3A__www.gams.com_-7Estefan&d=CwMFaQ&c=Ngd-ta5yRYsqeUsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=Krbuvr0BfvPSQKvQuiGrWIUfVTj8kednZerudAN3FWE&m=fWFE3-XGy0RrHtoT-9tu42xDFujqp0quF4dRDqzigTo&s=pYhWVu3U8HFur00P_HvCuo6CwyaLbdw6agTicgAvWoQ&e=>
>>>
>>
>>
>> _______________________________________________
>> Ipopt mailing list
>> Ipopt at list.coin-or.org
>> https://urldefense.proofpoint.com/v2/url?u=http-3A__list.coi
>> n-2Dor.org_mailman_listinfo_ipopt&d=CwICAg&c=Ngd-ta5yRYsqe
>> UsEDgxhcqsYYY1Xs5ogLxWPA_2Wlc4&r=Krbuvr0BfvPSQKvQuiGrWI
>> UfVTj8kednZerudAN3FWE&m=fWFE3-XGy0RrHtoT-9tu42xDFujqp0quF4dR
>> DqzigTo&s=6QkPtDoSYAmvd4vY3Xz5R459FRNnAPEZE5Kg_QcxhPA&e=
>>
>>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.coin-or.org/pipermail/ipopt/attachments/20160913/a2d95bed/attachment.html>
```