[Ipopt] optimality of solution

Andreas Waechter andreasw at watson.ibm.com
Mon Mar 14 10:03:58 EDT 2011

Hi Hossien,

The "lg(rg)" column in the Ipopt output tells you if the Hessian of the 
Lagrangian was modified during the step computation - if the reduced 
Hessian is not positive definite, then Ipopt adds a multiple of the 
identity to the Hessian matrix.  If you see a "-" (dash) in this column, 
it means that there was no modification (i.e., the reduced Hessian of the 
barrier problem is positive definite - up to numerical accuracy), and if 
there is a number, this is the log10 value of the regularization term.

So, if in the last iteration you see a "-" in that column, the reduced 
Hessian (of the barrier problem) is positive definite, and you most likely 
have a local minimum.  If there is a number, the final point might be 
stationary point that is not a local minimum (or the reduced Hessian is 
singular).  But all this is of course only computed within numerical 
accuracy, and this indicator might be wrong for badly scaled or 
ill-conditioned problems.

I hope this helps,


On Fri, 11 Mar 2011, Hossein Haghighat wrote:

> Hello,
> Does any body know how the optimality of the solution can be verified using
> IPOPT? I need to check the positive definiteness of the Hessian Matrix to
> make sure the solution is a local minimum not a saddle point or local
> maximum.
> -- 
> regards,
> Hossein.

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