<div dir="ltr"><div>Lasse,</div>
<div><br> </div>
<div>If you want to minimize a nonconvex function on set D, if your objective functional be strictly convex on D and set D be a closed convex set, then (under common conditions) your constrained optimization problem has a unique (global) minimizer. </div>
<div> </div>
<div>in this case, IPOPT will be enable to find your global solution.</div>
<div> </div>
<div>Hope this helps</div>
<div> </div>
<div>RT<br></div>
<div> </div>
<div><br> </div>
<div class="gmail_quote">On Wed, Nov 4, 2009 at 2:18 PM, Lasse Kliemann <span dir="ltr"><<a href="mailto:lasse-list-ipopt-2009@mail.plastictree.net" target="_blank">lasse-list-ipopt-2009@mail.plastictree.net</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0px 0px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">Hi Andreas,<br><br>thank you for the reply. My question is partly answered. Let me<br>emphasize one point that I am not sure about yet.<br>
<br>Given your formulation of the minimization problem below, define<br>the _feasible set_ by<br><br> F := { x | c(x) =0 and d(x) <= 0 }<br><br>Let c be linear and d be convex, so F is a convex set.<br><br>Let furthermore be the objective f convex on F. But we do not<br>
know how f behaves outside of F. Maybe we only know how f is<br>defined outside, and that it is differentiable. Assume this if<br>necessary.<br><br>A _local minimizer_ is a point x_0 in F such that there exists a<br>real number R>0 such that for all x in F<br>
<br> ||x-x_0|| <= R ---> f(x_0) <= f(x).<br><br>(I presume "first order optimal point" means exactly that?)<br><br>Now, it is clear that a local minimizer is a global minimizer,<br>i.e., a point x_0 such that f(x_0) = min {f(x) | x in F}. This<br>
follows from convexity of f on F.<br><br>But would Ipopt need convexity or some other nice properties<br>*outside the feasible set F*, in order to find a local minimizer?<br>I could imagine that the behavior of f close to the boundary of F<br>
might play at least a role. But I am not sure.<br><br>Thank you<br>Lasse<br><br>* Message by -Andreas Waechter- from Tue 2009-11-03:<br>> Hi Lasse,<br>><br>> If you write your optimization problem in the form<br>
><br>> min f(x)<br>> s.t. c(x) = 0<br>> d(x) <= 0<br>><br>> and f(x) and d(x) are convex, and c(x) is linear, then any first<br>> order optimal point (those are the points found my Ipopt) is a<br>
> global solution. Therefore, in that case Ipopt will find the global<br>> optimum.<br>><br>> However, if any of the functions is not convex (or the equality<br>> constraints c(x) are nonlinear), this is no longer true, and Ipopt<br>
> is guarateed by theory only to find a first order optimal point,<br>> which might not be the global solution. So, if you objective<br>> function is non-convex, Ipopt is not guaranteed to find a global<br>> solution. (Ipopt has measures that attempt to steer it towards a<br>
> local minimizer, though.)<br>><br>> But if your objective function is for some reason of a form so that<br>> every first order optimal point is a global solution (e.g., if it is<br>> pseudo-convex), then Ipopt will find the global solution. However,<br>
> this can be a rather special case.<br>><br>> Hope this helps,<br>><br>> Andreas<br>><br>> On Sat, 31 Oct 2009, Lasse Kliemann wrote:<br>><br>> >Greetings,<br>> ><br>> >I have questions concerning the behavior of Ipopt on convex<br>
> >problems. The documentation reads:<br>> ><br>> > It is important to keep in mind that the algorithm is only<br>> > trying to find a local minimizer of the problem; if the<br>> > problem is nonconvex, many stationary points with different<br>
> > objective function values might exist, and it depends on the<br>> > starting point and algorithmic choices which particular one<br>> > the method converges to.<br>> ><br>> >Now, assume I have a problem expressed with smooth functions<br>
> >(say, polynomials) and an objective function that is convex on<br>> >all R^n. Can I expect Ipopt to deliver a globally optimal<br>> >solution then (probably up to small inaccuracies)?<br>> ><br>
> >If so, my second question concerns the case when the objective<br>> >function is not convex on R^n, but at least the set K of feasible<br>> >points is convex and the objective function is convex on K. Can I<br>
> >expect Ipopt to deliver a globally optimal solution ("globally"<br>> >meaning: on K) then (probably up to small inaccuracies)?<br>> ><br>> >The latter occurs frequently in my application, so it would be<br>
> >interesting to know.<br>> ><br>> >By the way, thank you for your work on Ipopt!<br>> ><br>> >Kind regards<br>> >Lasse<br>> ><br><br>_______________________________________________<br>
Ipopt mailing list<br><a href="mailto:Ipopt@list.coin-or.org" target="_blank">Ipopt@list.coin-or.org</a><br><a href="http://list.coin-or.org/mailman/listinfo/ipopt" target="_blank">http://list.coin-or.org/mailman/listinfo/ipopt</a><br>
<br></blockquote></div><br></div>