Hey man I have seriously been enjoying your videos. I’ve been doing some work on these algorithms, focusing a bit more on the Newton or quasi Newton approaches for unconstrained problems. Turning constrained into unconstrained problems is just genius. I do have a question though. Numerically, it seems quadratic exterior penalty methods would be much better to work with. Introducing an ever sharper interior penalty method seems to be unstable. Especially if you are going to approximate the hessian, such a sharp kink in the barrier function seems … problematic. Any thoughts on how to fix that?
Both penalty methods are problematic in general and introduce ill-conditioning as the penalty parameter increases (or decreases for interior penalties). The main benefit of the interior penalty method is to introduce interior point methods, which behaves quite differently. But yes, if I was forced to use a penalty method than I would typically use an exterior penalty method unless other features of the problem dictated otherwise. As one example, if the analysis was undefined for infeasible values then an interior penalty would be preferable.
@@BYUFLOWLab wait, hold on, I thought interior penalty methods where basically identical to interior point methods? That’s what other literature seems to dictate. Am I mistaken? In any case, what would you suggest then? If penalty methods aren’t the way to go, what is the most modern of the algorithms to date? Lagrangian Multipliers and SQP seems to have quite some popularity
@@johnphilmore7269 As I understood from the video, Interior Penalty Methods is to add an interior log penalty to the objective function, whereas Interior Point Method is to introduce the slack variable to each inequality constraint, hence converting the inequality constraint to equality constraint.
@@hawkoli1987 ah thank you for the clear up. Honestly it has been a while since I was working on this. I think I eventually settled on a SQP algorithm. The interior barrier methods just … didn’t seem right to me. You needed to have a feasible solution anyway before you could apply it. I didn’t like that restrictiveness
Hey man I have seriously been enjoying your videos. I’ve been doing some work on these algorithms, focusing a bit more on the Newton or quasi Newton approaches for unconstrained problems. Turning constrained into unconstrained problems is just genius. I do have a question though. Numerically, it seems quadratic exterior penalty methods would be much better to work with. Introducing an ever sharper interior penalty method seems to be unstable. Especially if you are going to approximate the hessian, such a sharp kink in the barrier function seems … problematic. Any thoughts on how to fix that?
Both penalty methods are problematic in general and introduce ill-conditioning as the penalty parameter increases (or decreases for interior penalties). The main benefit of the interior penalty method is to introduce interior point methods, which behaves quite differently. But yes, if I was forced to use a penalty method than I would typically use an exterior penalty method unless other features of the problem dictated otherwise. As one example, if the analysis was undefined for infeasible values then an interior penalty would be preferable.
@@BYUFLOWLab wait, hold on, I thought interior penalty methods where basically identical to interior point methods? That’s what other literature seems to dictate. Am I mistaken?
In any case, what would you suggest then? If penalty methods aren’t the way to go, what is the most modern of the algorithms to date? Lagrangian Multipliers and SQP seems to have quite some popularity
@@johnphilmore7269 As I understood from the video, Interior Penalty Methods is to add an interior log penalty to the objective function, whereas Interior Point Method is to introduce the slack variable to each inequality constraint, hence converting the inequality constraint to equality constraint.
@@hawkoli1987 ah thank you for the clear up. Honestly it has been a while since I was working on this. I think I eventually settled on a SQP algorithm. The interior barrier methods just … didn’t seem right to me. You needed to have a feasible solution anyway before you could apply it. I didn’t like that restrictiveness