This paper presents a hybrid classical-quantum approach to solve mixed-integer linear programming (MILP) using neutral-atom quantum computations. We apply Benders decomposition (BD) to segment MILPs into a master problem (MP) and a subproblem, where the MP is addressed using a neutral-atom device, after being transformed into a quadratic unconstrained binary optimization (QUBO) model, with an automatized procedure. Our MILP to QUBO conversion tightens the upper bounds of the continuous variables involved, positively impacting the required qubit count, and the convergence of the algorithm. To solve the QUBO, we develop a heuristic for atom register embedding and apply a variational algorithm for pulse shaping. In addition, we implement a proof of concept that outperforms existing solutions. We also conduct preliminary numerical results: In a series of small MILP instances our algorithm identifies over 95% of feasible solutions of high quality, outperforming classical BD approaches where the MP is solved using simulated annealing.
This paper presents a hybrid classical-quantum approach to solve mixed-integer linear programming (MILP) using neutral-atom quantum computations. We apply Benders decomposition (BD) to segment MILPs into a master problem (MP) and a subproblem, where the MP is addressed using a neutral-atom device, after being transformed into a quadratic unconstrained binary optimization (QUBO) model, with an automatized procedure. Our MILP to QUBO conversion tightens the upper bounds of the continuous variables involved, positively impacting the required qubit count, and the convergence of the algorithm. To solve the QUBO, we develop a heuristic for atom register embedding and apply a variational algorithm for pulse shaping. In addition, we implement a proof of concept that outperforms existing solutions. We also conduct preliminary numerical results: In a series of small MILP instances our algorithm identifies over 95% of feasible solutions of high quality, outperforming classical BD approaches where the MP is solved using simulated annealing.
This paper presents key enhancements to our previous work [1] on a hybrid Benders decomposition (HBD) framework for solving mixed integer linear programs (MILPs). In our approach, the master problem is reformulated as a Quadratic Unconstrained Binary Optimization (QUBO) model and solved on a neutral-atom quantum processor using automated conversion techniques. Our enhancements address three critical challenges. First, to adapt to hardware constraints, we refine the QUBO formulation by tightening the bounds of continuous variables and employing an exponential encoding method that eliminates slack variables, thereby reducing the required qubit count. Second, to improve solution quality, we propose a robust feasibility cut generation method inspired by the L-shaped approach and implement a constructive penalty tuning mechanism that replaces manual settings. Third, to accelerate convergence, we introduce a multi-cut strategy that integrates multiple highdensity Benders cuts per iteration. Extensive numerical results demonstrate significant improvements compared to our previous approach: the feasibility rate increases from 68% to 100%, and the optimality rate rises from 52% to 86%. These advancements provide a solid foundation for future hybrid quantum-classical optimization solvers.
The Benders’ decomposition algorithm is a technique in mathematical programming for complex mixed-integer linear programming (MILP) problems with a particular block structure. The strategy of Benders’ decomposition can be described as a strategy of divide and conquer. The Benders’ decomposition algorithm has been employed in a variety of applications such as communication, networking, and machine learning. However, the master problem in Benders’ decomposition is still NP-hard, which motivates us to employ quantum computing. In the paper, we propose a hybrid quantum-classical Benders’ decomposition algorithm. We transfer the Benders’ decomposition’s master problem into the quadratic unconstrained binary optimization (QUBO) model and solve it by the state-of-the-art quantum annealer. Then, we analyze the computational results and discuss the feasibility of the proposed algorithm. Due to our reformulation in the master problem in Benders’ decomposition, our hybrid algorithm, which takes advantage of both classical and quantum computers, can guarantee the solution quality for solving MILP problems.
Mixed-integer linear programming problems are extensively used in industry for a wide range of optimization tasks. However, as they get larger, they present computational challenges for classical solvers within practical time limits. Quantum annealers can, in principle, accelerate the solution of problems formulated as quadratic unconstrained binary optimization instances, but their limited scale currently prevents achieving practical speedups. Quantum-classical algorithms have been proposed to take advantage of both paradigms and to allow current quantum computers to be used in larger problems. In this work, a hardware-agnostic Benders' decomposition algorithm and a series of enhancements with the goal of taking the most advantage of quantum computing are presented. The decomposition consists of a master problem with integer variables, which is reformulated as a quadratic unconstrained binary optimization problem and solved with a quantum annealer, and a linear subproblem solved by a classical computer. The enhancements consist, among others, of different embedding processes that substantially reduce the pre-processing time of the embedding computation without compromising solution quality, a conservative handling of cut constraints, and a stopping criterion that accounts for the limited size of current quantum computers and their heuristic nature. The proposed algorithm is benchmarked against classical approaches using a D-Wave quantum annealer for a scalable family of transmission network expansion planning problems.
Benders' decomposition (BD) algorithm constitutes a powerful mathematical programming method of solving mixed-integer linear programming (MILP) problems with a specific block structure. Nevertheless, BD still needs to solve an NP-hard quasi-integer programming master problem (MAP), which motivates us to harness the popular variational quantum algorithm (VQA) to assist BD. More specifically, we choose the popular quantum approximate optimization algorithm (QAOA) of the VQA family. We transfer the BD's MAP into a digital quantum circuit associated with a physically tangible problem-specific ansatz; and then solve it with the aid of a state-of-the-art digital quantum computer. Next, we evaluate the computational results and discuss the feasibility of the proposed algorithm. The hybrid approach advocated, which utilizes both classical and digital quantum computers, is capable of tackling many practical MILP problems in communication and networking, as demonstrated by a pair of case studies.
Mixed-integer linear programs (MILPs) are fundamental to power system operations but are challenging to solve due to their NP-hard nature. Quantum annealing, which seeks to find the ground state of an Ising model–a minimization of a quadratic binary function–offers potential speedups for solving combinatorial optimization problems. However, converting MILPs into quadratic unconstrained binary optimization (QUBO) problems often requires lots of slack variables, leading to significant qubit overhead and limiting the capability of quantum annealers. In this work, we propose an efficient formulation approach that generates a QUBO representation of a targeted MILP without introducing auxiliary binary variables. Geometrically, our method approximates the value function of the MILP with a quadratic binary function leveraging a simple convex optimization subroutine. We prove the existence of the QUBO representation, which preserves the MILP’s optimal solution and objective value with the same number of binary variables. The proposed approach is validated on several instances of the security-constrained unit commitment (SCUC) problem, with test cases featuring up to 288 binary variables, demonstrating its efficacy in reducing qubit overhead for quantum-classical hybrid Benders decomposition frameworks.
Edge computing holds significant potential to revolutionize computation by providing rapid and widely accessible services to numerous devices at the network edge. For service providers, efficient service placement and workload allocation are critical to minimizing operational costs while ensuring high-quality user experiences. In this work, we formulate this problem as a mixed-integer linear program (MILP), which is computationally challenging due to its NP-hard nature. To address this, we propose an innovative hybrid quantum-classical algorithm that integrates recent advances in quantum computing with the Benders decomposition method. Specifically, the MILP is decomposed into a master problem, which, after a series of transformations, can be solved using a quantum annealer, and a linear programming subproblem handled by a classical solver. These two problems are iteratively updated and solved, with information exchanged between the quantum and classical solvers, until convergence is achieved. Numerical experiments validate the effectiveness and computational advantages of the proposed method, demonstrating its practical applicability to real-world edge computing scenarios.