runs in a classical loop which checks for the number of selected assets $m(\Delta)$ in the best found solution with $\Delta=0$, then increases or decreases the individual desirability of the assets according to whether $m$ is larger or smaller than $M$ and runs again until $m(\Delta)=M$ for $\Delta=\Delta^\star$. This sort of hybridization scheme is not uncommon for quantum-assisted solvers [@tran2016hybrid; @venturelli2015quantum] and the number of expected rounds of runs should scale as $\propto \log_2(\Delta^\star)$ as per a binary search introducing a prefactor over the time-to-solution complexity which should stay manageable. Other hybrid approaches could also be put forward to deal with the constraint such as fixing some asset selections in pre-processing via sample persistence [@karimi2017boosting].

Per above arguments, in our benchmark case study in this work, we will focus on running unconstrained problems, setting $\Delta=0$. By design, the instance ensambles are such that the average number of assets in the ground state should be around $N/2$ (see Table \[tab:GreedySearchAndParams\]) corresponding to the largest combinatorial space by exhaustive enumeration of the possible solutions, so our tests will arguably be targeting among the most challenging instances of our parametrized model.\

Quantum Annealing Hybrid Solver {#sec:annealer}
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A QUBO problem can be easily 