I am new to optimisation and looking into VNS to attempt to apply a meta-heuristic to a MIQP in the form $$\max_x x^TQx$$ subject to: $x_i\in\{0,1\}$, $\sum_i x_i=C$, where $C$ is an integer.
I don't fully understand how to initialise the VNS algorithm i.e. how to select the set of neighborhood structures $$N_k(x) \ \textrm{for} \ k=1,...,k_{\textrm{max}}.$$
How do I select $N_k(x)$? Could it for example be the number of bits I am prepared to flip in $x$ (e.g. $k=1$ represents 1 bit, $k=2$ represents 2 bits etc.)? Or am I misunderstanding?
Many thanks!
You cannot flip an odd number of bits and preserve the equality constraint. Instead, let $k$ be the number of bits that flip from $0$ to $1$, and flip the same number from $1$ to $0$.