most recent 30 from http://mathoverflow.net 2013-05-21T07:01:46Z http://mathoverflow.net/feeds/question/94960 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94960/solving-a-particular-nonlinear-system-of-equalities Jennifer Gao 2012-04-23T16:00:42Z 2012-04-24T00:18:17Z

<p>How hard is it to solve a system of equalities of the form</p> <p>$a_{k1}x_1^k + \cdots + a_{kn}x_n^k = b_k$</p> <p>with $k$ ranging from $1$ to $m$? I realize that this is a non-convex system but it seems plausible that it might be tractable. If the theoretical complexity is bad, how might one go about finding a feasible solution to such a system in practice? In my case I have $m &lt; n \leq 10$. We also happen to know that $x_i \geq 0$, in case that helps. Other suggestions for tags are welcome.</p>

http://mathoverflow.net/questions/94960/solving-a-particular-nonlinear-system-of-equalities/94989#94989 Brian Borchers 2012-04-24T00:18:17Z 2012-04-24T00:18:17Z

<p>Your problem might be small enough that it is within the range of polynomial optimization techniques based on SDP relaxations of sums of squares problem. This has been implemented in software packages such as GLOPTIPOLY. See</p> <p><a href="http://homepages.laas.fr/henrion/software/gloptipoly3/" rel="nofollow">http://homepages.laas.fr/henrion/software/gloptipoly3/</a></p>