A method for solving a Lagrangian dual of a binary polynomially constrained polynomial programming problem by a quantum annealer, the method reducing sensitivity to errors of the quantum annealer and comprising: use of a processing unit for: obtaining a binary polynomially constrained polynomial programming problem; until a convergence is detected, iteratively, providing a set of Lagrange multipliers, providing an unconstrained binary quadratic programming problem representative of a Lagrangian relaxation of the binary polynomially constrained polynomial programming problem at the set of Lagrange multipliers provided, providing the unconstrained binary quadratic programming problem to a binary optimizer comprising the quantum annealer , obtaining from the quantum annealer at least one corresponding solution, using the at least one corresponding solution to generate a subgradient of a Lagrangian dual function of the binary polynomially constrained polynomial programming problem and to generate a new set of Lagrange multipliers based on the subgradient by the following equations: wherein λ i new , μ j new being a new Lagrange multiplier, λ i old , μ j old being a provided Lagrange multiplier, ∇ λi , ∇ μ j being a subgradient, α being a step size, L ( λ, μ) being a Lagrangian function, g i being an equality constraint, h j being an inequality constraint, x being a solution of the unconstrained binary quadratic programming problem, m being a number of equality constraints, I being a number of inequality constraints; and providing all corresponding best-known primal-dual pairs of the Lagrangian dual of the binary polynomially constrained polynomial programming problem and best-known feasible solutions of the binary polynomially constrained polynomial programming problem after the convergence. obtaining a binary polynomially constrained polynomial programming problem; until a convergence is detected, iteratively, providing a set of Lagrange multipliers, providing an unconstrained binary quadratic programming problem representative of a Lagrangian relaxation of the binary polynomially constrained polynomial programming problem at the set of Lagrange multipliers provided, providing the unconstrained binary quadratic programming problem to a binary optimizer comprising the quantum annealer , obtaining from the quantum annealer at least one corresponding solution, using the at least one corresponding solution to generate a subgradient of a Lagrangian dual function of the binary polynomially constrained polynomial programming problem and to generate a new set of Lagrange multipliers based on the subgradient by the following equations: λ i new , μ j new being a new Lagrange multiplier, λ i old , μ j old being a provided Lagrange multiplier, ∇ λi , ∇ μ j being a subgradient, α being a step size, L ( λ, μ) being a Lagrangian function, g i being an equality constraint, h j being an inequality constraint, x being a solution of the unconstrained binary quadratic programming problem, m being a number of equality constraints, I being a number of inequality constraints; and providing all corresponding best-known primal-dual pairs of the Lagrangian dual of the binary polynomially constrained polynomial programming problem and best-known feasible solutions of the binary polynomially constrained polynomial programming problem after the convergence. <EOS>