Say I have a vector of integers ${\bf a} = [a_1,\cdots,a_N]^T$ and an optimizations problem I would like to find a solution (or approximation) in this set.
Assume for every number, a probability distribution ${\bf p} = [p_1,p_2,\cdots,p_N]^T$, so that $$\sum_{k=1}^{N} p_k = {\bf 1}^T{\bf p} = 1$$
Then we can calculate expectation:
$$E[{\bf a}] = \sum_{k=1}^{N}p_ka_k = {\bf a}^T{\bf p}$$
Now we can enforce the first one with a squares cost term: $$\|{\bf 1}^T{\bf p}-1\|_2^2$$
We can use the second one to steer the solution towards the correct mean value:
$$\|{\bf a}^T{\bf p}-\mu\|_2^2$$
Now what remains is some kind of variance or entropy minimization. Ideally we want a solution that tells us with $100\%$ certainty exactly what each integer variable should be, this corresponds to variance $0$ and entropy $0$.
Own approach:
Re-weighted Tikhonov regularization $${\bf \Lambda }_{n}= \text{diag}( f({\bf v}_{n-1}) )$$
Where $f$ is some function which maps $1$ to small costs and $0$ to large costs (and negative numbers to even larger costs).
How feasible is this approach?
Which other methods can be used?
Is there some kind of guarantee of convergence?