Performance enhancing of hybrid quantum-classical Benders approach for MILP optimization

TL;DR

This work tackles the hardness of MILP in large-scale optimization by introducing a hardware‑agnostic hybrid quantum–classical Benders decomposition (BDQA) that solves the integer master as a QUBO on a quantum annealer while the linear subproblem is handled classically. Key innovations include precomputed embeddings to drastically reduce embedding preprocessing time, conservative handling of cut constraints, and a practical stopping criterion tailored to current quantum hardware limits, demonstrated on a transmission network expansion planning (TNEP) problem. Empirical results show that precomputed embeddings can reduce total runtime by about an order of magnitude without sacrificing solution quality, and that the BDQA framework remains competitive for small and medium instances while highlighting the need for larger quantum devices to scale further. The study advances the integration of quantum resources into large-scale optimization in energy systems, outlining concrete enhancements and future directions for cut generation, embedding optimization, and non‑convex or time‑constrained problem classes.

Abstract

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.

Figures (11)

• Figure 1: Diagram of the Benders’ Decomposition algorithm. The squared area corresponds to the quantum subroutine of the algorithm.
• Figure 2: Average objective values compared to the state-of-the-art solution and the 5% acceptance margin (shaded area). Error bars represent the standard deviation across 10% successful runs of configuration: (RUNS=100, READS=100).
• Figure 3: Success rate as function of the number of buses of configuration: (RUNS=100, READS=100).
• Figure 4: Average number of iterations required for convergence as function of the number of buses. Error bars represent the standard deviation across 10% successful runs of configuration: (RUNS=100, READS=100).
• Figure 5: Average (last) QUBO size of each problem instance as function of the number of buses. Error bars represent the standard deviation across 10% successful runs of configuration: (RUNS=100, READS=100).