# Minimize pure quadratic subject to linear equality constraint

The problem to solve is

$\min_x \sum_i \sum_j C_{ij} x_i x_j$ $\,$ s.t. $\sum_i x_i = 1$.

for $C$ a symmetric and positive semidefinite matrix. Or equivalently

$\min_x x'Cx$ $\,$ s.t. $x'e = 1$.

This feels similar to a quadratic minimization onto the positive simplex (for a particular norm induced by C), although here the individual $x_i$ might take negative values. As already stated in How can I minimize a quadratic on the unit simplex? such problem can be solved with general quadratic programming methods, which also applies here. However, it seems that for this particular case a closed form solution can be found as

$x_i = \frac{\sum_j C_{ij}^{-1}}{\sum_k \sum_j C_{kj}^{-1}}$,

as stated (without proof) in the paper "When Networks Disagree: Ensemble Methods for Hybrid Neural Networks", section 4: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.3857

How can this result be obtained?

And furthermore, would still be possible to find a closed form solution if positivity constraints are enforced as $x \geq 0$, recovering the positive simplex problem but for a pure symmetric quadratic objective?

Using Lagrange multipliers, $x$ is a solution of your problem if there is $\mu$ such that $$2Cx + \mu e =0, \ x^Te =1.$$ Solving the first equation for $x$ gives $$x = -\frac\mu2 C^{-1}e.$$ However, $x^Te=1$ is required, so $$\mu = -2 (e^TC^{-1}e)^{-1}.$$ And we get $$x= \frac{C^{-1}e}{e^TC^{-1}e}.$$