I have the following integer optimisation problem, and I wonder whether it can be reformulated as a conic program that can be solved with, e.g., Mosek. Suppose the $n$-dimensional vectors $a, b$ and $c$ are fixed. I want to solve

$$ \max~(a \cdot x)\prod_{i}b_i^{x_i} + (a \cdot y)\prod_{i}b_i^{y_i} $$ $$ \text{s.t. }\ x, y \in \{0,1\}^n, $$ $$ x_i + y_i \leq c_i,\ \forall i \in [n], $$ $$ \sum_{i} x_i \leq 10, \sum_{i} y_i \leq 10. $$

My first attempt at solving this problem was to reformulate the problem so that it can be solved by SCIP. Unfortunately, SCIP takes far too long on instances that I need to solve.

Focusing on the simpler objective $\max~(a \cdot x)\prod_{i}b_i^{x_i}$ (so ignoring $y$ entirely), I managed to formulate the problem as a mixed-integer conic program, and Mosek was able to solve this much more quickly. My hope is that reformulating the original general problem will yield a similar speedup. But I've not been able to achieve this so far - any insights would be greatly appreciated. Alternatively, if there is some other clever reformulation, I would also be very interested to hear about this.

Here's my conic reformulation for the simpler objective, in case it helps. I use the fact that $\log((a \cdot x)\prod_{i}b_i^{x_i}) = \log(a\cdot x) + x_i \log b_i$.

$$ \max \ w $$ $$ \text{s.t. }w = l + x \cdot \log(b), $$ $$ l \leq \log(z), $$ $$ z = a \cdot x, $$ $$ x_i \leq c_i, \forall i \in [n], $$ $$ \sum_{i}x_i \leq 10. $$ In particular, the constraint $l \leq \log(z)$ corresponds to the conic constraint $ (z,1,l) \in K_{exp}$. Is it possible to somehow extend this reformulation to the original problem?