Karush Khan Tucker Gradient Conditions

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Hi, initially I cut the photo off the book called Nonlinear-and-Mixed-Integer-Optimization-Fundamentals-and-Applications-Topics-in-Chemical-Engineering and page 120.

My question is: in the GREEN box it says we have to use KKT gradient conditions with respect to a_i and as a result of that we obtain the set in RED box. You see the sum of u_i equals to 1 BUTTT when I take gradient conditions into account with respect to a_i, I find that all individual u_i must be 1. FOR INSTANCE say that Lagrangian function is sum of a_i + sum of u_i*[ g_i () - a_i ], and apply KKT gradient conditions to Lagrangian function with respect to a_1. Derivative of sum of a_i with respect to "a_1" is just " 1 " and
derivative of sum of u_i*[ g_i () - a_i ] with respect to "a_1" equals to just " -u_1", then summation of them must equal to "0", then all individual u_i must be 1 INSTEAD of summation of u_i equals 1??
Could you help me about that?
 

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This seems to be a very specialized areas on math and we don't always have experts that can answer your questions.

In any event, I did a google search and found these references that may help you find your answer:

https://www.google.com/search?newwi....0.96.96.1......0....1j2..gws-wiz.OwByeZ0gc_4

There are numerous videos too:

https://www.google.com/search?q=Kar...k9ndAhWi6IMKHU66BckQ_AUIDygC&biw=1391&bih=911

In particular the APMonitor seems to have a series of videos on this:

Initially thanks for your return. But I would like to express that I have already done the necessary calculations and shown them in my post 1. I just can not reach the same point like the mentioned book. Could you help me about that? I always find all individual u_i equals 1 instead of sum of u_i equals 1?
 
  • #4
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Sorry I can't help here my expertise is limited. I take it the videos didn't help?
 
  • #5
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Sorry I can't help here my expertise is limited. I take it the videos didn't help?
The videos contain what I have already learnt. But I appreciate it. I just would like to see whether to make some mistakes?
 
  • #6
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HI again, I would like to say I know KKT karush khan tucker conditions and tried to take derivative of mentioned equation in previous posts. But I think I am making a mistake while taking gradients. I find all individual u_i must be 1 but truly as you can see in screenshot sum of u_i must be 1. Could you help me notice my mistake?
 
  • #7
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I know this has to be frustrating, but the problem here is that we don't always have someone versed in your problem area to help. I tried by posting the videos. I've checked around and no one immediately comes to mind. You may need to talk with any profs you know versed in this area.

You could also try some other online sites but you may experience the same problem with not enough subject experts.

Lastly, you may just have to accept the result for now and come back to it later when you've got more experience with KKT.
 

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