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.
You have a cardinality constrained QP, known to be hard (intractable in the general case). You can formulate it as a mixed-integer QP and solve it globally, or apply some heuristic and hope for the best. Searching for the topic will lead you to a wealth of references and material.