2:51 Big picture 5:23 Composition 10:14 Vector composition 13:49 Minimization 22:39 Perspective 27:17 The conjugate function 34:09 Examples(of conjugate) 37:50 Quasiconvex functions 47:18 internal rate of return 53:20 Properties 56:41 Log-concave and log-convex functions 1:02:12 Properties of log-concave functions 1:06:26 Consequences of integration property 1:07:48 example: yield function 1:12:01 Convexity with respect to generalized inequalities "Every now and then accidentally you will manufacture an acceptable point" ~ Boyd
Prof. Boyd's sense of humour has me cracking up every now and then. Never happended in real life lectures (also never stayed awake in real life lectures before)
After computing the conjugate of -log x, at about 36:30, it would have been awesome if they had shown that the conjugate of the conjugate is again -log x .
1:50 Why should a symmentric matrix be added to a Positive Semi Defenite (PSD) Matrix to stay in the subspace? Why is V not PSD? How to generally find the type of matrix that should be added to a given group to stay in the subspace? For example if we have SE3 what should I add to it (what is a good candidate for V) so that I stay in the subspace? Thank You
A symmetric matrix is added because you want it to stay in symmetric (and not PSD) subspace. The just comes from the formulation of the problem. Domain of X is PSD space but domain of function f is symmetric space.
Remember that all this results are based on euclidean linear algebra and calculus, the direction idea stated that you should change from R^n->R function to a R->R function that alows all possible lines over your set S, but to consider all posible lines in the set you need that x+vt belongs, where x is in your set but where your direction is arbitrary (any vector in R^n). Symetric matrices can be encoded S^n can be enconded in R^m, where m=n(n+1)/2 (because we only the diagonal and half the matrix) and there's where your direction come from, to answer the question in the video you can't ensure the line stays in the set, you can assume it stays as you want to prove convexity over it, but you equally have to state that v is any vector and no PD, and to answer your final question (that in the real world doesn't matter, you have to ensure your matrices are PD to apply results like log det convexity) for a good candidate you don't want a specific V, but mostly a very small t (that doesn't introduce rounding errors tough) as PD are an open set of symmetric matrices, and then there are neighborhoods around x, so that all matrices in all directions are PD
People who have done this video series, how much of the functions and the results obtained until now are used in the subsequent lessons ? I am unable to follow all of it due to mathematical rigor involved.
53:00 To have IRR(x) >= R, for all r less than R, we should have PV(x,r) NOT EQUAL TO zero. In the lecture it says greater than equal to. I am sure the equality should hold only at r = R. A stronger way of saying that would be PV strictly > 0. But What I do not understand is why PV can't be negative. Thanks
At 33:00 we define the convex envelope of f as the function f^env whose epigraph is the convex hull of the epigraph of f. But I see a little catch: in this way, we are defining the epigraph of f^env, not f^env itself. Is f^env uniquely defined by its epigraph?
Yes, a convex function is uniquely defined by its epigraph. The epigraph of a function is the set of points lying on or above the graph of the function. For a convex function, its epigraph forms a convex set. Conversely, if we have a convex set, we can define a convex function associated with it. Specifically, for any convex set S, there exists a unique convex function f(x) such that the epigraph of f(x) is precisely S. This relationship between convex functions and their epigraphs is one-to-one, meaning that each convex function has a unique epigraph, and each convex set has a unique associated convex function. Therefore, the epigraph of a convex function fully characterizes the function itself, and we can determine a convex function uniquely from its epigraph.
Well, that's just the formula for calculating the current value of an asset on which you are paying r% interest rate for i years for example.. nothing to do with optimization I guess
2:51 Big picture
5:23 Composition
10:14 Vector composition
13:49 Minimization
22:39 Perspective
27:17 The conjugate function
34:09 Examples(of conjugate)
37:50 Quasiconvex functions
47:18 internal rate of return
53:20 Properties
56:41 Log-concave and log-convex functions
1:02:12 Properties of log-concave functions
1:06:26 Consequences of integration property
1:07:48 example: yield function
1:12:01 Convexity with respect to generalized inequalities
"Every now and then accidentally you will manufacture an acceptable point" ~ Boyd
Thanks a lot man, really grateful to you.
ty based shivanand
谢谢你课代表🤣
Prof. Boyd's sense of humour has me cracking up every now and then. Never happended in real life lectures (also never stayed awake in real life lectures before)
Thanks for your generosity in sharing the lectures and text books on line!
After computing the conjugate of -log x, at about 36:30, it would have been awesome if they had shown that the conjugate of the conjugate is again -log x .
I mean that's trivial. The conjugate operation is an involution. QED.
Coolest prof ever!!!
1:50
Why should a symmentric matrix be added to a Positive Semi Defenite (PSD) Matrix to stay in the subspace? Why is V not PSD? How to generally find the type of matrix that should be added to a given group to stay in the subspace? For example if we have SE3 what should I add to it (what is a good candidate for V) so that I stay in the subspace?
Thank You
because the difference of 2 PSD matrices is a symmetric matrix which is not necessarily PSD however.
A symmetric matrix is added because you want it to stay in symmetric (and not PSD) subspace. The just comes from the formulation of the problem. Domain of X is PSD space but domain of function f is symmetric space.
Remember that all this results are based on euclidean linear algebra and calculus, the direction idea stated that you should change from R^n->R function to a R->R function that alows all possible lines over your set S, but to consider all posible lines in the set you need that x+vt belongs, where x is in your set but where your direction is arbitrary (any vector in R^n).
Symetric matrices can be encoded S^n can be enconded in R^m, where m=n(n+1)/2 (because we only the diagonal and half the matrix) and there's where your direction come from, to answer the question in the video you can't ensure the line stays in the set, you can assume it stays as you want to prove convexity over it, but you equally have to state that v is any vector and no PD,
and to answer your final question (that in the real world doesn't matter, you have to ensure your matrices are PD to apply results like log det convexity) for a good candidate you don't want a specific V, but mostly a very small t (that doesn't introduce rounding errors tough) as PD are an open set of symmetric matrices, and then there are neighborhoods around x, so that all matrices in all directions are PD
People who have done this video series, how much of the functions and the results obtained until now are used in the subsequent lessons ? I am unable to follow all of it due to mathematical rigor involved.
Thank you professor Boyd
53:00
To have IRR(x) >= R, for all r less than R, we should have PV(x,r) NOT EQUAL TO zero. In the lecture it says greater than equal to. I am sure the equality should hold only at r = R. A stronger way of saying that would be PV strictly > 0. But What I do not understand is why PV can't be negative.
Thanks
At 33:00 we define the convex envelope of f as the function f^env whose epigraph is the convex hull of the epigraph of f. But I see a little catch: in this way, we are defining the epigraph of f^env, not f^env itself. Is f^env uniquely defined by its epigraph?
Yes, a convex function is uniquely defined by its epigraph.
The epigraph of a function is the set of points lying on or above the graph of the function. For a convex function, its epigraph forms a convex set.
Conversely, if we have a convex set, we can define a convex function associated with it. Specifically, for any convex set S, there exists a unique convex function f(x) such that the epigraph of f(x) is precisely S.
This relationship between convex functions and their epigraphs is one-to-one, meaning that each convex function has a unique epigraph, and each convex set has a unique associated convex function.
Therefore, the epigraph of a convex function fully characterizes the function itself, and we can determine a convex function uniquely from its epigraph.
In the book the Schur complement is defined as: S=C - B' A-1 B. Here, at around 20:18 it's defined as S=A - B C-1 B'. Anyone know why?
The equation changes if you minimize differently. The book minimizes x whereas in this lecture he minimizes y. There is a better explanation in A.5.5.
Delsin Menolascino As Patrick said, Schur has multiple forms. Wikipedia also does a good job summarizing Boyd's work in the appendix.
I suspect the present value function is defined that way BECAUSE it is quasi convex, and it's not a coincidence.
Well, that's just the formula for calculating the current value of an asset on which you are paying r% interest rate for i years for example.. nothing to do with optimization I guess
38:46 "They make little diagrams.....", careful you will be in trouble .........hahahahaaaa....love the professor...
Why does he keep saying something is increasing what in fact is non-descreasing?
@laputahayom Convex Optimization - Boyd and Vandenberghe
How can I get the home works for this course?
2:46 Sounded like that student ended up hearing that *every* "type" of matrix is closed under addition.
I don't understand at 8:51 why we cannot apply the composition theorem
That is because your function is not increasing anymore since for z
I have a problem. In 40:24, why is beta sublevel set convex?
The beta sublevel set is S={x\in\R: x
Great!
Convex Optimization
Stephen Boyd and Lieven Vandenberghe
Cambridge University Press
i need solution manual of Additional Exercises for Convex Optimization by Stephen Boyd " anyone please
@@ismailelezi can you share the link again (nqs ke mundesi :)
can anyone tell me why sqrt abs x is quasiconvex?
looks like concave all the way to me...
Not concave at 0. Also, informally, a concave function should be "peaking" only once. Sqrt abs x is peaking twice
@1:03:27 🤯
What is a positive semidefinite matrix ?
It's a matrix with eigen values 0 or positive, but not negative.
"Keeps otherwise dangerous people off the streets" LOL
LOL...quasi convex inventor is fuming...