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

[Victor Wu]

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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