Timeline for An optimization problem on the sphere

8 events

when toggle format	what		by	license	comment
Mar 4, 2014 at 23:57	history	edited	Turbo		edited tags
Mar 3, 2014 at 15:31	comment	added	Alex Degtyarev		Of course, it's $(\delta_i^2-r)$. I don't know how to handle the integral problem. Over $\mathbb{R}$, you need the points in $X$ where the two gradients span a plane containing $U([1,\ldots,1])$: $$\lambda[v_1,\ldots,v_n]+\mu[(\delta_1^2-r)v_1,\ldots,(\delta_n^2-r)v_n]=U([1,\dots,1])$$ for some $\lambda,\mu\in\mathbb{R}$, but I do not see an easy way to solve this system of a lot of quadratic equations (in $v_1,\ldots,v_n,\lambda,\mu$).
Mar 3, 2014 at 15:25	history	edited	Turbo	CC BY-SA 3.0	added 73 characters in body
Mar 3, 2014 at 11:55	history	edited	Turbo	CC BY-SA 3.0	added 147 characters in body
Mar 3, 2014 at 11:37	history	edited	Turbo	CC BY-SA 3.0	added 42 characters in body
Mar 3, 2014 at 11:33	comment	added	Turbo		Could you develop it as an answer? I believe the terms should be $(\delta_i^2-r)v_i^2$.
Mar 3, 2014 at 10:35	comment	added	Alex Degtyarev		First, disregard $U$; you get the quadric cone $\sum_i(\delta_i-r)v_i^2=0$, where $\delta_i$ are the eigenvalues of $D$. Intersect it with the sphere to get a codimension $2$ (typically) variety, say $X$. You want $U^{-1}(X)$. Then, use Lagrange multipliers?
Mar 3, 2014 at 10:22	history	asked	Turbo	CC BY-SA 3.0