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May
1
comment why not handle box-constraints with a transformation
Hi Apurv: Thanks for your reply. Unfortunately, I don't follow it because for the problem I described, lagrange multipliers ( i.e: $\lambda$ ) are not involved.
Apr
23
comment intuition behind subspace of $R^n$
@Blind: Hi: I can email you the page from Byrne's text that has the statements of the two theorems and their proofs. Or I can also latex it in another thread if you want them that way. Thanks and no rush.
Apr
23
comment intuition behind subspace of $R^n$
@Blind: Hi. Although I checked the answer, I went back to my text and the author has something slightly different than you. He has the same theorem as you wrote for the subspace ( the one you titled: projection onto a halfspace but that's probably a typo ) but he claims that it holds for a linear manifold rather than a subspace. Note that the author also has a seperate but related theorem for the case of subspace. The result is the same as the one that you wrote for the subspace except that he has $y$ instead of $y - P_{c}(x)$ in the second term. ???? Thanks for any clarification on this.
Apr
22
comment intuition behind subspace of $R^n$
@Blind: Thank you. That was great. I followed every step and the definitions are now in sync with what Charles Byrne has in his book. Now, I can go back to it and continue reading and wait for confusion to come. When it does, I will write again to this list. It's amazing. Thanks again.
Apr
22
accepted intuition behind subspace of $R^n$
Apr
20
comment intuition behind subspace of $R^n$
Previous should have read : " A subset S of $R^{n}$". Sorry for confusion.
Apr
20
comment intuition behind subspace of $R^n$
I have to leave now but here is how subspace is defined in "A First Course In Optimization:": A subset of $R^{n}$ is a subspace is for every x and y in S and scalars $\alpha$ and $\beta $, the linear combination $\alpha * x + \beta * y $ again belongs to S. But that sounds similar to your linear manifold definition. Thanks.
Apr
20
comment intuition behind subspace of $R^n$
In Prop 2, I don't see "Hence, M is a subspace". I stopped there and will try to understand Prop 3 later. Thanks for any clearing my "denseness". P.S: The way Byrne's defines subspace looks quite similar to how you define linear manifold. I'll fight through both and try to connect the dots.
Apr
20
comment intuition behind subspace of $R^n$
Little O said it very simply and it made sense to me but I want to try and understand teh mathematics of it which is what you wrote. I didn't even go over propistion 3 yet, but I'm even confused on one piece in rop1 and one piece in prop 2. In prop 1, I don't see why "Certainly, O belongs to M".
Apr
20
comment intuition behind subspace of $R^n$
@little0: Your simple explanation is clear and helpful. I still need to read Blind's explanation carefully. I meant that c is a "point" belong to C where C is a convex set and some subspace of $R^{n}$. Sorry for the lack of clarity. I asked another question before this in a different thread and this was kind o a continuation of that one.
Apr
20
comment intuition behind subspace of $R^n$
Hi Blind: I'm confident that I will accept it but I need to go through it carefully and I haven't had the chance to yet. Your thorough and helpful answer is appreciated.
Apr
19
comment intuition behind subspace of $R^n$
Thanks. I feel like you're providing a private course in linear algebra for convex optimization. I will print out and read carefully as I always do. As I mentioned earlier, do you obtain these various proofs-relations from a book or notes or just from your brain ? Also, I don't have Boyd's text at the moment ( well. I know I bought it in the past but I can't find it which is quite annoying ) but might that be a text for intuition-figures etc. Thanks again and I'll check it when I understand it. That could take some time.
Apr
19
asked intuition behind subspace of $R^n$
Apr
19
comment proof that length of difference of projections implies equality of length of normals
@Blind: That was an amazing answer. I don't know what else to say besides thanks. It took me a while to understand the algebra and the various relations but you explained it all beautfully. I'm going to ask another question ( more of a confirmation ) using a different thread. Feel free to read that one also. Thanks again.
Apr
19
accepted proof that length of difference of projections implies equality of length of normals
Apr
18
comment proof that length of difference of projections implies equality of length of normals
@Blind: Thank you but I need to understand it first and then I will definitely accept it. But I don't follow all the steps yet. I'll get back to you if I can't figure them ( the ones I don't follow yet ) out myself.
Apr
18
comment proof that length of difference of projections implies equality of length of normals
Thank you. I will print out and go over carefully and then check the answer. It's appreciated.
Apr
18
comment proof that length of difference of projections implies equality of length of normals
@Batman: Thank you for the hint. I will try that approach. It's appreciated.
Apr
18
comment proof that length of difference of projections implies equality of length of normals
Also, if anyone knows of a book or notes where topics like the one in this proof are explained in detail for "dummies", ( i.e: not rockefellar or lemarchal ) and where there might be figures to aid in the understanding, it's appreciated.
Apr
18
comment proof that length of difference of projections implies equality of length of normals
@Robert: Thanks. I should have said projection onto a closed convex set. Apologies for that also.