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

Ipopt User ipoptuser at gmail.com
Mon Sep 12 13:24:54 EDT 2016

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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