Mod-01 Lec-12 Convex Optimization

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Thank you very much Professor Dutta for the lecture. I have been having great fun following the journey. I just wish to ask a question: I see when you are using the separation theorem to show that S is in fact equal to the normal cone of C at xbar,  you put zero in the middle of the inequalities rather than some unknown alfa. Can I do so in other proofs as well if the set in question is a cone? In lecture 14, when we prove that K is K itself, why did we not use zero as alfa and make the proof simpler?

lf
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In 2:49, why do we have that optimization? What is the linkage between normal cone definition and this problem?

tsznamchan