<div dir="ltr">If you want a guarantee for optimality, you need a KKT point. You can easily verify if the limit point satisfies the KKT conditions.</div><div class="gmail_extra"><br><div class="gmail_quote">On Tue, Sep 13, 2016 at 1:10 PM, Victor Wu <span dir="ltr"><<a href="mailto:vwwu@umich.edu" target="_blank">vwwu@umich.edu</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">I will be using a limited memory approximation. Also, it may be difficult to find a feasible starting solution.<div><br></div><div>It seems that a convex extension of the objective outside the feasible region would be a safer way to guarantee global optimality?</div></div><div class="HOEnZb"><div class="h5"><div class="gmail_extra"><br><div class="gmail_quote">On Mon, Sep 12, 2016 at 1:24 PM, Ipopt User <span dir="ltr"><<a href="mailto:ipoptuser@gmail.com" target="_blank">ipoptuser@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">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).</div><div class="gmail_extra"><br><div class="gmail_quote"><div><div>On Mon, Sep 12, 2016 at 10:58 AM, Victor Wu <span dir="ltr"><<a href="mailto:vwwu@umich.edu" target="_blank">vwwu@umich.edu</a>></span> wrote:<br></div></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div><div><div dir="ltr">Hi Stefan,<div><br></div><div>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)?<div><br></div><div>Victor</div></div></div><div class="gmail_extra"><br><div class="gmail_quote"><span>On Mon, Sep 12, 2016 at 10:47 AM, Stefan Vigerske <span dir="ltr"><<a href="mailto:stefan@math.hu-berlin.de" target="_blank">stefan@math.hu-berlin.de</a>></span> wrote:<br></span><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><span>Hi,<br>
<br>
if you provide a feasible starting point, I would imagine that nonconvexity of the objective function outside of the feasible area doesn't matter.<br>
<br>
Stefan<div><div><br>
<br>
<br>
On 09/12/2016 04:36 PM, Victor Wu wrote:<br>
</div></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div><div>
Hello,<br>
<br>
I have a nonlinear problem where the objective function is non-convex in<br>
general, but is convex over the (convex) feasible region of interest (thus,<br>
theoretically I have a convex problem). Do you have any recommendations on<br>
how I should represent the objective (i.e., write the objective in its<br>
original non-convex form and enforce my feasible region or do something<br>
else) in IpOpt? My concern is whether IpOpt would reach the<br>
*theoretically* global minimizer and if it depends on my implementation.<br>
<br>
Thank you,<br>
Victor Wu<br>
<br>
<br>
<br></div></div>
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