Maximise this quadaratic form, subject to these constraints.

[Phillips101]

Question:

Given a1+a2+a3+...+an=0 and a1^2+a2^2+...+an^2=1, (all real numbers)

find the maximal value of a1*a2+a2*a3+...+an*a1

Thoughts so far:

I've treated the expression as a combination of n variables and differentiated - when it came to putting the constraints in it got to be a hideous mess.

It is easily factorisable as 0.5*( (a1+a2)^2 + (a2+a3)^2 +...+ (an+a1)^2 ) -1, but then maximising the inside is just as hard.

Help would be appreciated! (Also, sorry about the lack of LATEX knowhow).

 

[trambolin]

You can write the original constraints as
$$\max \ \ a^T\begin{pmatrix} 0 &1 &\ldots &\ldots 0\\ 0 &0 &1 &\ldots 0\\ \vdots \\1 &0 &\ldots &0\end{pmatrix}a$$
subject to
$$\begin{pmatrix} a &\mathbf{1} \end{pmatrix}^T a = \begin{pmatrix}1 \\0\end{pmatrix}$$

and try to fit the problem to linear or quadratic programming frame

 

[Phillips101]

I know of linear programming, but I have no idea how to go about solving that using it unfortunately. And nor do I think I'm expected to know, this is a linear algebra question, which is the annoying thing...

And I'd never heard of or encountered quadratic programming before.