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

Ipopt User ipoptuser at gmail.com
Tue Sep 13 13:38:06 EDT 2016

```If you want a guarantee for optimality, you need a KKT point. You can
easily verify if the limit point satisfies the KKT conditions.

On Tue, Sep 13, 2016 at 1:10 PM, Victor Wu <vwwu at umich.edu> wrote:

> 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,
>>>
>>> Thank you for your quick response!  A follow up question: if I start
>>> with 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
>>>>>
>>>>>
>>>>>
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>>>>
>>>> --
>>>> http://www.gams.com/~stefan
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>>
>
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