1:59 Affine Set 3:06 Convex Set 5:28 Convex combination and convex hull 9:40 Convex cones 10:52 Hyperplanes and halfspaces 14:03 Euclidean balls and ellipsoids 15:49 Norm balls and norm cones 20:12 Polyhedra 23:19 Positive semidefinite cone 30:34 Operations that preserve convexity 34:42 Intersection 36:27 Affine function 43:39 Perspective and linear-fractional function 47:32 Generalized inequalities 56:53 Minimum and minimal elements 1:01:55 Separating hyperplane theorem 1:03:34 Supporting hyperplane theorem 1:07:24 Dual cones and generalized inequalities 1:11:24 Minimum and minimal elements via dual inequalities 1:13:20 optimal production frontier
Thank you so much! What you did right here reduced the suffering of many and probably freed up space and time for those people to go on and reduce yet others suffering! May the echoes of this goodness remain good and reach back to help you some day.
The explanation in this video is great and understandable if listener has a very solid linear algebra background. It was difficult to me a few years ago, but after I seriously reviewed my linear algebra, the whole video makes more sense to me.
I really appreciated the demonstrations. I do not have a pure math background and seeing the demonstrations over and over again make it easier to see convexity everywhere.
As foretold in a Jedi prophecy, the Chosen One would bring about the destruction of the *Sets* and the restoration of balance in the Force. Great lecture though!
Wow, I expected some student as a guest lecturer, but this guy teaches a good lecture! Does Stanford provide some basic training for anyone eligible to teach?
Around 36min, jacob makes the claim that cos(t), cos(2t), ... are all convex therefore a linear combination of them is also convex. However, I believe this is false. Consider the line y = 1 - (2/pi)t. Then, for example, the intersection is defined as cos(t) = 1 - (2/pi)t. The t-coordinate of two of the solutions are t = 0 and t = pi/2 and t = pi. However, cos(t) > 1 - (2/pi)t for 0 < t < pi/2 and cos(t) < 1 - (2/pi)t for pi/2 < t < pi. Just want to make sure...
I feel like the example given at 49:59 is not actually a convex cone: a convex cone must be a convex set, but this clearly is not. If we take a point from the first quadrant and other from the third, then the conic combination could lie in the second quadrant, which is outside of this cone.
Correct. That was an example of a cone that is not convex just to demonstrate that not all cones are convex, and what we usually visualize as cones are the special ones, i.e., proper convex cones.
It is the extension to inequalities applied to matrices. The sign means that the matrix in question is classified as negative definite, or in other words, all of its eigenvalues are real and strictly negative.
It is said that non negative orthant is a proper cone. In R2 it turns out to be the first quadrant. My question is can we say other quadrants or in general orthants are also proper cones? If not what is the special property exhibited by the first quadrant or the non negative orthant which is not exhibited by others due to which the non negative orthant turns out to be a proper cone?
One example to describe why it is needed to restrain the convex cone to be pointed could be the space V is actually a cone and a convex cone. But it provides no useful information on the "ordering" as it contains a line. so it is not a proper cone. another incorrect point in the clip is that the scaler for a cone is positive not nonnegative. PS: I found on wiki that a cone is pointed when {0} is not included, which is confusing.
I think textbook has an error for perspective function. Textbook says on page 40 that mu = (theta * x_n+1) / (theta * x_n+1 + (1-theta) * y_n+1) But when I multiply this into muP(x) + (1-mu)P(y) the result is not the same
This lecture is only good as a supplement to the book. He absolutely flies through about 38 pages of fairly dense material and examples. There's little to no rigorous development of the topics and some he just completely skips.
At what level is this course taught in Stanford. I am an undergrad sophomore and I am being taught this in our curriculum. Couldnt understand anything well at all.
@kazvah If you turn on the speech recognition, the transcriber has some problems with this, e.g. "half spaces are convicts" :-) Nevertheless a great lecture by Jacob
I think he might have showed a not very good example for generalized inequality at 56:40, because the difference between the two vector will not be in the proper cone.
Does anyone understand why t has to range from 0 to 1 to make the set {x in R^m | x1 + x2t + ... + xnt^(n-1), t in [0,1]} a convex set and proper cone? Could t also theoretically range from say [0,5] or some other arbitrary range? Why specifically is it "t in [0,1]"?
He messes up "proper cone" where he tries to give a counterexample of a non-pointed cone. The correct counter-example is x >= 0, y>=0, and z arbitrary. It contains the line x=0, y=0, and z arbitrary.
I am reading this book and I very much appreciate the book being available for free and the quality of its content. But it would also help, a lot, if the lectures weren't garbage. "Go read the book" is not an acceptable answer to every fucking question.
I wanted to make some mathematica code for the cone duality Block[{e, f}, e = FullSimplify[ Evaluate[ c > 0 && d > 0 && a . {c, d} < 0 && b . {c, d} > 0 /. a -> {1, -2} /. b -> {1, -1} /. c -> a /. d -> b]]; f = ToString[e]; With[{te1 = e, te2 = f}, RegionPlot[{ToString[FullSimplify[te1 && (x a + y b > 0)]] == te2, te1 /. a -> x /. b -> y}, {x, -2, 2}, {y, -2, 2}, Axes -> True, AxesLabel -> {x, y}, AxesOrigin -> {0, 0}, Ticks -> {{-2, -1, 0, 1, 2}, {-2, -1, 0, 1, 2}}, PlotPoints -> 100]] ]
I think you got confused with vectors and x-y co-ordinates. In vector form, x_vec = θ(x1_vec) + (1 − θ)x2_vec can be re-written of the form transpose(a_vec)*x_vec=constant; so it is a line. To simply see that. Put x_vec = [x;y], x1_vec = [x1;y1] and x2_vec = [x2;y2] in the vector equation, you can re write it to y = x(y1-y2)/(x1-x2)+y2-x2(y1-y2)/(x1-x2). this is a line in 2-D of the form y = mx+c
According to Boyd : "the symbol denotes vector inequality or componentwise inequality in dimension m : u ≺v means u i ≤ v i for i = 1, . . . , m." For those who're wondering
Generalized inequality: x ≺k y if y-x in k. For example, real number positive number R+: if x1 in R+, x2 in R+, then if x1 ≺R+ x2, implies x2-x1 in R+. This means ≺R+ is pretty much < sign on real positive numbers, because x1 < x2 implies x2-x1 > 0 (x2-x1 is still a positive number in R+).
I think textbook has an error for perspective function. Textbook says on page 40 that mu = (theta * x_n+1) / (theta * x_n+1 + (1-theta) * y_n+1) But when I multiply this into muP(x) + (1-mu)P(y) the result is not the same
1:59 Affine Set
3:06 Convex Set
5:28 Convex combination and convex hull
9:40 Convex cones
10:52 Hyperplanes and halfspaces
14:03 Euclidean balls and ellipsoids
15:49 Norm balls and norm cones
20:12 Polyhedra
23:19 Positive semidefinite cone
30:34 Operations that preserve convexity
34:42 Intersection
36:27 Affine function
43:39 Perspective and linear-fractional function
47:32 Generalized inequalities
56:53 Minimum and minimal elements
1:01:55 Separating hyperplane theorem
1:03:34 Supporting hyperplane theorem
1:07:24 Dual cones and generalized inequalities
1:11:24 Minimum and minimal elements via dual inequalities
1:13:20 optimal production frontier
Doing God's work here!
Thank you so much!
What you did right here reduced the suffering of many and probably freed up space and time for those people to go on and reduce yet others suffering!
May the echoes of this goodness remain good and reach back to help you some day.
@@aname5241 Thank you for your very kind words.
The explanation in this video is great and understandable if listener has a very solid linear algebra background. It was difficult to me a few years ago, but after I seriously reviewed my linear algebra, the whole video makes more sense to me.
I really appreciated the demonstrations. I do not have a pure math background and seeing the demonstrations over and over again make it easier to see convexity everywhere.
The dual part is pretty unclear, for example, it says that x for all 2 norm
The dotted lines in the figures (40:00 - 47:00) show the boundary of the domains of the function and its inverse respectively.
it is very easy to follow. just replace all sound "i" with "e"
He does not have accent. He simply shifts the alphabeth.
As foretold in a Jedi prophecy, the Chosen One would bring about the destruction of the *Sets* and the restoration of balance in the Force. Great lecture though!
lol
The line segment is derived from the positivity constraint on the denominator. x1 + x2 + 1 = 0, so it defines the domain of the function.
Wow, I expected some student as a guest lecturer, but this guy teaches a good lecture! Does Stanford provide some basic training for anyone eligible to teach?
+Rob Romijnders I agree. Only his accent is a bit annoying... "sets"
thats the bést
Around 36min, jacob makes the claim that cos(t), cos(2t), ... are all convex therefore a linear combination of them is also convex. However, I believe this is false. Consider the line y = 1 - (2/pi)t. Then, for example, the intersection is defined as cos(t) = 1 - (2/pi)t. The t-coordinate of two of the solutions are t = 0 and t = pi/2 and t = pi. However, cos(t) > 1 - (2/pi)t for 0 < t < pi/2 and cos(t) < 1 - (2/pi)t for pi/2 < t < pi. Just want to make sure...
at the part about positive semidefinite cone,I think a positive definite matrix must be symmetric
Yes, I did a double take on that. PD matrices must be symmetric
I feel like the example given at 49:59 is not actually a convex cone: a convex cone must be a convex set, but this clearly is not. If we take a point from the first quadrant and other from the third, then the conic combination could lie in the second quadrant, which is outside of this cone.
Correct. That was an example of a cone that is not convex just to demonstrate that not all cones are convex, and what we usually visualize as cones are the special ones, i.e., proper convex cones.
at the end, *puts pen together, throws it* what a boss
Ugh spoilers
It is the extension to inequalities applied to matrices. The sign means that the matrix in question is classified as negative definite, or in other words, all of its eigenvalues are real and strictly negative.
Optimistic discussion. Hoping for more.
This lecture is all about causing a lot of damage!
Damn think this guys speaks better than the actual lecturer
Great lecture, easy to understand, thanks Jacob.
1:16:35 "that's how it is done."
It is said that non negative orthant is a proper cone. In R2 it turns out to be the first quadrant. My question is can we say other quadrants or in general orthants are also proper cones? If not what is the special property exhibited by the first quadrant or the non negative orthant which is not exhibited by others due to which the non negative orthant turns out to be a proper cone?
One example to describe why it is needed to restrain the convex cone to be pointed could be the space V is actually a cone and a convex cone. But it provides no useful information on the "ordering" as it contains a line. so it is not a proper cone. another incorrect point in the clip is that the scaler for a cone is positive not nonnegative. PS: I found on wiki that a cone is pointed when {0} is not included, which is confusing.
I thought control system was tough.
I think textbook has an error for perspective function.
Textbook says on page 40 that mu = (theta * x_n+1) / (theta * x_n+1 + (1-theta) * y_n+1)
But when I multiply this into muP(x) + (1-mu)P(y) the result is not the same
Very good professor here
EE 364A Course Website: link is broken.
This lecture is only good as a supplement to the book. He absolutely flies through about 38 pages of fairly dense material and examples. There's little to no rigorous development of the topics and some he just completely skips.
The dotted line at 46:40 is the equation x1+x2+1=0
Matrix analysis for Statistics (James R. Schott) の Separating hyperplane theorem の理解にとても有用でした。
At what level is this course taught in Stanford. I am an undergrad sophomore and I am being taught this in our curriculum. Couldnt understand anything well at all.
what is this course belongs to the EE deparment?
TYPO, it should be WHY
Anyone can send link to the h.w please?
43:12, in the hyperbolic cone example, f(x)'f(x) = x'half(p)'half(p)'x + (c'x)^2 = x'Px + (c'x)^2
Good stuff. Very clear. Interesting accent.
@kazvah
If you turn on the speech recognition, the transcriber has some problems with this, e.g. "half spaces are convicts" :-)
Nevertheless a great lecture by Jacob
I think he might have showed a not very good example for generalized inequality at 56:40, because the difference between the two vector will not be in the proper cone.
康神！
nice lecture, Mr. John Wick
Chopinm4n: the curly less than means less than in the PSD cone sense (rather than the matrix elementwise sense)
correction: `...of the three solutions are...' instead of `...of two of the solutions are...'
Are these slide downloadable?
I think he got confused with minimum and minimal on the last page
Does anyone understand why t has to range from 0 to 1 to make the set {x in R^m | x1 + x2t + ... + xnt^(n-1), t in [0,1]} a convex set and proper cone? Could t also theoretically range from say [0,5] or some other arbitrary range? Why specifically is it "t in [0,1]"?
I think when you write transpose here, you usually mean dot product?
What is the book?
He messes up "proper cone" where he tries to give a counterexample of a non-pointed cone. The correct counter-example is x >= 0, y>=0, and z arbitrary. It contains the line x=0, y=0, and z arbitrary.
anon31415 correct. it was corrected at 53:00
I am reading this book and I very much appreciate the book being available for free and the quality of its content. But it would also help, a lot, if the lectures weren't garbage. "Go read the book" is not an acceptable answer to every fucking question.
which book????
Khushboo Mawatwal convex optimization by Boyd (and a second author)
I wanted to make some mathematica code for the cone duality
Block[{e, f},
e = FullSimplify[
Evaluate[
c > 0 && d > 0 && a . {c, d} < 0 && b . {c, d} > 0 /.
a -> {1, -2} /. b -> {1, -1} /. c -> a /. d -> b]];
f = ToString[e];
With[{te1 = e, te2 = f},
RegionPlot[{ToString[FullSimplify[te1 && (x a + y b > 0)]] == te2,
te1 /. a -> x /. b -> y}, {x, -2, 2}, {y, -2, 2}, Axes -> True,
AxesLabel -> {x, y}, AxesOrigin -> {0, 0},
Ticks -> {{-2, -1, 0, 1, 2}, {-2, -1, 0, 1, 2}}, PlotPoints -> 100]]
]
From where i can download the chapter ? or any book
web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf it is chapter 2
Can anyone give me a link which proves that y = θx1 + (1 − θ)x2 represents the points on a line. I have looked up on web, but could not find this.
ax + by = c is an equation of a straight line. In this case, a=θ and b=1-θ.
I think you got confused with vectors and x-y co-ordinates. In vector form, x_vec = θ(x1_vec) + (1 − θ)x2_vec can be re-written of the form transpose(a_vec)*x_vec=constant; so it is a line. To simply see that. Put x_vec = [x;y], x1_vec = [x1;y1] and x2_vec = [x2;y2] in the vector equation, you can re write it to y = x(y1-y2)/(x1-x2)+y2-x2(y1-y2)/(x1-x2). this is a line in 2-D of the form y = mx+c
book name?
What a great lecture
Thanks!
Can anyone tell me chapter 2 of which book he referring to?
Book : web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf
Slide : web.stanford.edu/~boyd/cvxbook/bv_cvxslides.pdf
Is lecture 1 available ?
ua-cam.com/users/view_play_list?p=3940DD956CDF0622
Does anyone know what that curly less than sign ( ≺ )means specifically? Is it literally just less than, or does it have any other meaning?
According to Boyd : "the symbol denotes vector inequality or componentwise inequality in dimension m :
u ≺v means u i ≤ v i for i = 1, . . . , m."
For those who're wondering
Generalized inequality: x ≺k y if y-x in k. For example, real number positive number R+: if x1 in R+, x2 in R+, then if x1 ≺R+ x2, implies x2-x1 in R+. This means ≺R+ is pretty much < sign on real positive numbers, because x1 < x2 implies x2-x1 > 0 (x2-x1 is still a positive number in R+).
I think of it as a generalized symbol for an arbitrary ordering.
this particular lecture is going too fast
normally I wouldn't stand this accent, especially on the "e" like sets. But he is just sooooo HOT! started to like his accent a little bit.
lol
fuck u women, ruining engineering and mathematics.
None of these things I'm watching has nothing to do with, my question!
I asked a question about my STIMULUS CHECK. Where Is the answer?
Good lecture, but the following bugs me:
"sit" (set)
"projict" (project)
etc.
The man is just somehow repeating Prof. Boyd's textbook.
The lecture guy's accent is so hot
that got hard fast
Is this guy related to Keanu Reeves?
From my point of view, this lecturer is not vert clear... He didn't show a good understanding towards the stuff he is talking about.
哈哈哈银老板也在看CVX啊
He Yin he is only the replacement
nice accent better than some profs from poland or india
although it is still not easy to understand all his words
According to your logic, his accent is better than some profs from China too?
seeDOTstanfordDOTedu/see/materials/lsocoee364a/assignmentsDOTaspx
what kind of English is this ???
He is really hot~
Actually he's really cool. Is it possible to be both hot and cool? I think not!
dude is so so steaming hot~!
......
So difficult for me to understand his accent. Have to pause and play back every several minutes. poor me!
Very bad!
I think textbook has an error for perspective function.
Textbook says on page 40 that mu = (theta * x_n+1) / (theta * x_n+1 + (1-theta) * y_n+1)
But when I multiply this into muP(x) + (1-mu)P(y) the result is not the same