A method is disclosed for solving the Lagrangian dual of a constrained binary quadratic programming problem. The method comprises obtaining a constrained quadratic binary programming problem; until a convergence is detected, iteratively, performing a Lagrangian relaxation of the constrained quadratic binary programming problem to provide an unconstrained quadratic binary programming problem, providing the unconstrained quadratic binary programming problem to a quantum annealer, obtaining from the quantum annealer at least one corresponding solution, using the at least one corresponding solution to generate a new approximation for the Lagrangian dual bound; and providing a corresponding solution to the Lagrangian dual of the constrained binary quadratic programming problem after convergence.