Scheduling multiple federated learning (FL) models within a distributed network, especially in large-scale scenarios, poses significant challenges since it involves solving NP-hard mixed-integer nonlinear programming (MINLP) problems. However, it's imperative to optimize participant selection and learning rate determination for these FL models to avoid excessive training costs and prevent resource contention. While some existing methods focus solely on optimizing a single global FL model, others struggle to achieve optimal solutions as the problem grows more complex. In this paper, exploiting the potential of quantum computing, we introduce the Hybrid Quantum-Classical Benders' Decomposition (HQCBD) algorithm to effectively tackle the joint MINLP optimization problem for multi-model FL training. HQCBD combines quantum and classical computing to solve the joint participant selection and learning scheduling problem. It decomposes the optimization problem into a master problem with binary variables and small subproblems with continuous variables, then leverages the strengths of both quantum and classical computing to solve them respectively and iteratively. Furthermore, we propose the Hybrid Quantum-Classical Multiple-cuts Benders' Decomposition (MBD) algorithm, which utilizes the inherent capabilities of quantum algorithms to produce multiple cuts in each round, to speed up the proposed HQCBD algorithm. Extensive simulation on the commercial quantum annealing machine demonstrates the effectiveness and robustness of the proposed methods (both HQCBD and MBD), with improvements of up to 70.3% in iterations and 81% in computation time over the classical Benders' decomposition algorithm on classical CPUs, even at modest scales.
Scheduling multiple federated learning (FL) models within a distributed network, especially in large-scale scenarios, poses significant challenges since it involves solving NP-hard mixed-integer nonlinear programming (MINLP) problems. However, it's imperative to optimize participant selection and learning rate determination for these FL models to avoid excessive training costs and prevent resource contention. While some existing methods focus solely on optimizing a single global FL model, others struggle to achieve optimal solutions as the problem grows more complex. In this paper, exploiting the potential of quantum computing, we introduce the Hybrid Quantum-Classical Benders' Decomposition (HQCBD) algorithm to effectively tackle the joint MINLP optimization problem for multi-model FL training. HQCBD combines quantum and classical computing to solve the joint participant selection and learning scheduling problem. It decomposes the optimization problem into a master problem with binary variables and small subproblems with continuous variables, then leverages the strengths of both quantum and classical computing to solve them respectively and iteratively. Furthermore, we propose the Hybrid Quantum-Classical Multiple-cuts Benders' Decomposition (MBD) algorithm, which utilizes the inherent capabilities of quantum algorithms to produce multiple cuts in each round, to speed up the proposed HQCBD algorithm. Extensive simulation on the commercial quantum annealing machine demonstrates the effectiveness and robustness of the proposed methods (both HQCBD and MBD), with improvements of up to 70.3% in iterations and 81% in computation time over the classical Benders' decomposition algorithm on classical CPUs, even at modest scales.
The scheduling problem for federated learning (FL) with multiple models in a distributed network is challenging, as it involves NP-hard mixed-integer nonlinear programming. Moreover, it requires optimal participant selection and learning rate determination among multiple FL models to avoid high training costs and resource competition. To overcome those chal-lenges, in literature the Benders' decomposition algorithm (BD) can deal with mixed integer problems, however, it still suffers from limited scalability. To address this issue, in this paper, we present the Hybrid Quantum-Classical Benders' Decomposition (HQCBD) algorithm, which combines the power of quantum and classical computing to solve the joint participant selection and learning scheduling problem in multi-model FL. HQCBD decomposes the optimization problem into a master problem with binary variables and small subproblems with continuous variables. This collaboration maximizes the potential of both quantum and classical computing, and optimizes the complex joint optimization problem. Simulation on the commercial D-Wave quantum annealing machine demonstrates the effectiveness and robustness of the proposed method, with up to 18% improvement of iterations and 81% improvement of computation time over BD algorithm on classical CPUs even at small scales.
• We propose a new hybrid quantum-classical algorithm for solving the UC problem. • The hybrid model successfully optimizes 24-hour UC systems from 10 to 1000 units. • The model outperforms classical metaheuristics (GA and SA) and classical MINLP. • This represents an improvement over previous QCUC methods, a step to real-world use. Resource scheduling is critical in many industries, especially in power systems where the Unit Commitment (UC) problem determines the on/off status and output levels of generators under physical and economic constraints. Traditional exact methods, such as Branch‑and‑Bound, Branch‑and‑Cut, dynamic programming and mixed‑integer linear programming (MILP), remain the backbone of UC solution techniques, but they often rely on linear approximations or exhaustive search, leading to high computational burdens as system size grows. Metaheuristic approaches, such as genetic algorithms, particle swarm optimization, and other evolutionary methods, have been explored to mitigate this complexity; however, they typically lack optimality guarantees, exhibit sensitivity to initial conditions, and can become prohibitively time‑consuming for large‑scale systems. In this paper, we introduce a quantum‑classical hybrid algorithm for UC, and other resource scheduling problems by extension, that leverages Benders decomposition to decouple binary commitment decisions from continuous economic dispatch. The binary “master problem” is formulated as a quadratic unconstrained binary optimization (QUBO) model and solved on a quantum annealer. The continuous “subproblem,” which minimizes generation costs, with Lagrangian cuts feeding back to the master until convergence. We evaluate our hybrid framework on systems scaled from 10 to 1,000 generation units. Compared against a classical mixed‑integer nonlinear programming (MINLP) baseline, the hybrid algorithm achieves a consistently lower computation‑time growth rate and maintains an absolute optimality gap below 1.63%. These results demonstrate that integrating quantum annealing within a hybrid quantum-classical Benders decomposition loop can significantly accelerate large‑scale resource scheduling without sacrificing solution quality, pointing toward a viable path for addressing the escalating complexity of modern power grids.
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.
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.
During recent years, quantum computers have received increasing attention, primarily due to their ability to significantly increase computational performance for specific problems. Computational performance could be improved for mathematical optimization by quantum annealers. This special type of quantum computer can solve quadratic unconstrained binary optimization problems. However, multi-energy systems optimization commonly involves integer and continuous decision variables. Due to their mixed-integer problem structure, quantum annealers cannot be directly used for multi-energy system optimization. To solve multi-energy system optimization problems, we present a hybrid Benders decomposition approach combining optimization on quantum and classical computers. In our approach, the quantum computer solves the master problem, which involves only the integer variables from the original energy system optimization problem. The subproblem includes the continuous variables and is solved by a classical computer. For better performance, we apply improvement techniques to the Benders decomposition. We test the approach on a case study to design a cost-optimal multi-energy system. While we provide a proof of concept that our Benders decomposition approach is applicable for the design of multi-energy systems, the computational time is still higher than for approaches using classical computers only. We therefore estimate the potential improvement of our approach to be expected for larger and fault-tolerant quantum computers.
Bilevel programming problems (BLPPs) frequently appear in hierarchical decision-making contexts, including supply chain network design, critical infrastructure defense, and energy market equilibrium. These problems are NP-hard, posing significant computational challenges due to their nested structure. Classical exact solvers often struggle when the upper-level (leader) objective involves dense quadratic interactions, as linearization techniques typically induce an exponential increase in variables and constraints. Conversely, Quantum Annealing (QA) approaches offer potential advantages for quadratic optimization but suffer from resolution errors and qubit scarcity when handling continuous lower-level variables directly. To address these limitations, this study proposes a Hybrid Quantum Benders Decomposition (HQBD) framework. The method decomposes the Quadratic-Linear BLPP into a quantum master problem, which manages discrete combinatorial decisions with quadratic terms, and a classical subproblem, which handles continuous resource allocation. Numerical experiments on highly frustrated 3D Torus topologies demonstrate the framework’s algorithmic scalability up toN= 343 variables (2343combinatorial states), a scale where state-of-the-art classical exact solvers (Gurobi MIQP) trigger a 600-second timeout due to dense quadratic constraints. Furthermore, to mitigate early-stage barren plateaus commonly found in Benders-QUBO integrations, we introduce a “Relaxed Resource Strategy.” To fully address existing literature gaps and align with quantum computing standards, this revised study introduces a comprehensive sensitivity analysis of the relaxation parameters and expands the scalability benchmarking across five spatial dimensions. Statistical robustness analysis over 20 independent trials reveals that the optimized strategy achieves instantaneous convergence (Mean = 1.0 iteration), completely bypassing the exponential classical complexity wall.