# Quadratic programming with constrained number of free variables

I started with a (positive-definite) quadratic programming problem subject only to a single equality constraint. i.e.

$$f(x)=x^{T}Qx+c^{T}x$$ $$s.t. x_1+x_2+x_3+...+x_n=y$$

I now have to find the optimal solution subject to varying $m<n$ of the parameters with all other variables fixed to be zero. I'm well aware that I can now reduce the dimensionality of the problem to a $Q$-matrix of order $m$ and that's not the issue.

The issue is choosing the 'optimal' set of $m$ parameters that best minimize the objective function. Essentially, I have to find the optimal optimization from the $^nC_m$ possible choices of free variables to play with.

I don't really know how to classify this problem let alone solve it. Assume $m$ is typically 7 and $n$ is typically 100 so about 16 billion problems via brute force :S

Any idea how to tackle this? Help, thoughts much obliged.