# Rahul Narain

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bio website location age 28 member for 3 years, 4 months seen 8 mins ago profile views 3,946

Almost all of the questions on the front page these days are homework questions or textbook exercises. I think I'll be spending a lot less time here.

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 2d comment Shortest closed loop containing all extreme points of a convex set Flipping intersecting edges is a standard lemma for showing that the optimal travelling salesman tour is always a simple polygon. 2d comment Shortest closed loop containing all extreme points of a convex set Perhaps the case of infinitely many vertices can be attacked through a limiting process. 2d comment Shortest closed loop containing all extreme points of a convex set For the case of finely many vertices, I believe this follows from (i) for a given ordering, the shortest path is a polygon, (ii) a self-intersecting polygon can be made shorter by "flipping" the intersecting edges, and (iii) any nonconvex polygon on a convex set of vertices is self-intersecting. Dec17 comment Distance between point and a spiral You seem to be assuming that the closest point on the spiral lies along the line from the origin to the point in question, which is not correct. Dec17 comment What is matrix inequality such as $A>0$ or $A\succ 0$? "How do I know which definition of inequality people are meaning?" You can't from the tiny snippets you are posting. Usually they will have defined what they mean earlier in the text. Dec16 comment Are solutions to optimization problems with smooth, continuous, and strictly concave objective functions and linear constraints always unique? Linear constraints (whether equality or inequality) are always convex. I don't know what a spherical objective function is. If you mean something like $x^2+y^2+z^2$, then yes. Dec16 comment Are solutions to optimization problems with smooth, continuous, and strictly concave objective functions and linear constraints always unique? However, a concave function may not have a unique minimum, for example $\min -x^2$ for $-1\le x\le 1$. Dec16 comment Are solutions to optimization problems with smooth, continuous, and strictly concave objective functions and linear constraints always unique? A strictly convex function on a convex domain has a unique minimum (unless it is unbounded below). A strictly concave function on a convex domain has a unique maximum (unless it is unbounded above). See e.g. en.wikipedia.org/wiki/Convex_optimization Dec16 comment How to solve a linear program with OR constraints Do you mean that (constraint 1 is true for all $i$) or (constraint 2 is true for all $i$)? Or do you mean that (constraint 1 is true or constraint 2 is true) for all $i$? I think you mean the latter, but it would be good to edit the question to make it perfectly clear. Dec14 comment What is the idea behind the $^2$ in the mean squared error? Probably the same idea as in Motivation behind standard deviation? Dec14 comment Which is bigger, $1+3\sqrt{2}$ or $3\sqrt{3}$? Now you just have to compare $8$ to $2\sqrt{18}$, which you can do by squaring. (Or observe that $8=2\sqrt{16}$.) Dec14 comment Are vertices with no edges part of any graphs? If you store vertices as endpoints of edges, then you will get multiple copies of each vertex, which is usually not what you want. Dec14 comment How to determine if an array of digits is random? Dec13 comment optimization area problem Please see How to ask a homework question? You should improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. Dec13 comment Can a derivative be undefined at a local maxima? Consider $f(x)=-|x|$. Dec13 revised Flow Graphs: Why do you need the symmetry property of a graph? added 123 characters in body Dec12 comment minimizing water slide speed. Dec11 comment proving not all integers are sums of 3 squares? The square of an odd number, modulo $4$, is always $1$ and never $3$. Dec11 comment How to determine that a surface is symmetric @apt1002: That's a good idea. We can take only the highest-order terms and then, at least in 2D, replace $x$ and $y$ with $\cos\theta$ and $\sin\theta$ to get a function $g(\theta)$. The value of $\theta$ about which $g$ is symmetric is the only possible direction of the symmetry axis. (Note that $2\tan^{-1}(1/2) = \tan^{-1}(4/3)$.) Dec10 revised Defining a graph as G=(V,E) — how to interpret the notation? added 6 characters in body