I'd like a good way to solve an optimization problem I came across. It's a constrained knapsack problem: I want to find integers
$1=a_1\le a_2\le\cdots\le a_t$
$a_1+a_2+\cdots+a_t=N$
with $t$ as small as possible such that (with the $S_i\subseteq S={a_1,a_2,\ldots,a_t}$)
$\sum_{e\in S_i}e=C_i$
for a certain fixed collection $C_1 = 1, C_2 = 4,\ldots,C_k=N.$
What software exists for this sort of problem? I feel like writing my own program from the ground up would be the wrong approach.
I've been toying with COIN-OR and such but they seem set up for very different sorts of problems.
Ideally, I'd solve an even more complicated problem where $C_1$ and $C_k$ must match exactly and the other sums must be within a fixed range depending on the values of the neighboring C-values... but I don't imagine this is feasible?