QUBO Sampling for Mixed Binary Quadratic Programming without Continuous Variable Binarization
Quantum annealing and related combinatorial optimization methods typically accept quadratic unconstrained binary optimization (QUBO) problems as input, whereas many practical models include constraints and continuous variables. Standard QUBO conversions discretize continuous variables, increasing the binary dimension and often making feasible low-energy states harder to sample. We develop a Lagrange-multiplier method for a separable class of mixed-binary quadratic programs (MBQPs) without discretizing continuous variables. The method is based on a finite-temperature free-energy formulation and uses QUBO sampling for the binary sector. At fixed Lagrange multipliers, the continuous sector is integrated out analytically and enters only the multiplier update, leaving a QUBO over the original binary variables. We evaluate the method on the continuous relaxation of the quadratic $p$-median problem. Compared with a penalty-based QUBO formulation, it generates feasible solutions more reliably. At an appropriate inverse temperature, its conditional relative error is comparable to that of local search for small instances and often lower for the larger tested instances. In the time-to-target comparisons, the proposed method maintains high target-reached rates for stringent fixed-accuracy targets. For the larger instances within the tested range, it also reaches the prescribed targets faster than a commercial mixed-integer optimization solver.