Convex optimization book - solution - exercise - 2.3 - midpoint convexity

[@darrirro]

8:20 z2 = x/2 + x/4 + y/4 btw no y/2 , but thanks for the explanation

[@mathelecs]

You are right. That should be y/4 not y/2. Thank you for pointing that out!

[@hank-pm1rd]

07:51 z1 is the midpoint of x and y, why do you say "if z1 is closer to x or closer to y"

[@mathelecs]

That's very good question. I misspoke. I should have said if z is closer to x or y. Every time that we find a midpoint from two end points, we compare z to those end points. Hope this clears the confusion.

[@hank-pm1rd]

@@mathelecs thank u very much for clarification!

[@mathelecs]

@@hank-pm1rd My pleasure. Thank you for watching.

[@孙天成-g8z]

May I ask why Lamda k for k to the infinity then it becomes Lamda? is limZk for k to the infinite means that to cut the line for the infinite times so that we can get any point on the line?

[@mathelecs]

@孙天成: Absolutely! lambda_k is the sum of the coefficient of x. When you let k go to infinity, this sum is convergent and converges to a point between 0 and one. That number is the limit of the sum which is denoted by Lambda. Hopefully, this answers your question. Do not hesitate to ask follow up questions. I will be happy to answer them.

[@passager683]

@@mathelecs Great sequential approach to the problem, thank you. I’m a little bit suspicious about the lambda_k sequence. How can we be sure it converges to THE lambda set forth in the beginning ? Unless any lambda between 0 and 1 can be written as such, which is not necessarily true if lambda is irrational for example (a finite sum of rationals is a rational). Thank you again, and good luck.

[@mathelecs]

@@passager683 It is my pleasure! Honestly, that is a very great question. I do not know the answer. At the time I was creating the video, I was trying to give a proof that makes the proof in the solution clear (egrcc.github.io/docs/math/cvxbook-solutions.pdf). I would be happy to know the answer. Thank you for reaching out.

[@passager683]

@@mathelecs It all makes sense now, That formula is actually the partial sum of order k of the BINARY representation of theta as an INFINITE series of zeros and ones (the c_k), which is legitimate as ALL real numbers (namely those between 0 and 1) can be written as such. A great gem in an unexpected place. Keep up the good work !

[@mathelecs]

@@passager683 I so glad you that cleared up the confusion. I appreciate your kind words! I will do my best to create as much as videos I can.