Combinatorial optimization with a smooth and convex objective function arises naturally in applications such as discrete mean-variance portfolio optimization, where assets must be traded in integer quantities. Although optimal solutions to the associated smooth problem can be computed efficiently, existing adiabatic quantum optimization methods cannot leverage this information. Moreover, while various warm-starting strategies have been proposed for gate-based quantum optimization, none of them explicitly integrate insights from the relaxed continuous solution into the QUBO formulation. In this work, a novel approach is introduced that restricts the search space to discrete solutions in the vicinity of the continuous optimum by constructing a compact Hilbert space, thereby reducing the number of required qubits. Experiments on software solvers and a D-Wave Advantage quantum annealer demonstrate that our method outperforms state-of-the-art techniques.