Parallel splitting method for large-scale quadratic programs

TL;DR

This work tackles the challenge of solving large-scale quadratic programs by introducing SPLIT, a matheuristic that partitions the problem into $K$ subgraphs with local variables $X_k$ and local fields $D_k$, solving subproblems in parallel while iteratively accounting for inter-subproblem interactions through correction terms $Δ_k$. The framework provides a flexible, solver-agnostic approach that can incorporate hard constraints or penalties and leverages spectral clustering for partitioning, enabling near real-time solutions or scalability beyond current quantum hardware limits. Empirical results on MaxCut and Antenna Placement demonstrate substantial speedups and high-quality solutions up to tens of thousands of variables, validating SPLIT's applicability to industrial-scale problems and its potential to bridge classical and quantum-inspired optimization. Future directions include extending to non-binary variables, refining constraint handling, and exploring dynamic reallocation of variables among subproblems to further boost efficiency and solution quality.

Abstract

Current algorithms for large-scale industrial optimization problems typically face a trade-off: they either require exponential time to reach optimal solutions, or employ problem-specific heuristics. To overcome these limitations, we introduce SPLIT, a general-purpose quantum-inspired framework for decomposing large-scale quadratic programs into smaller subproblems, which are then solved in parallel. SPLIT accounts for cross-interactions between subproblems, which are usually neglected in other decomposition techniques. The SPLIT framework can integrate generic subproblem solvers, ranging from standard branch-and-bound methods to quantum optimization algorithms. We demonstrate its effectiveness through comparisons with commercial solvers on the MaxCut and Antenna Placement Problems, with up to 20,000 decision variables. Our results show that SPLIT is capable of providing drastic reductions in computational time, while delivering high-quality solutions. In these regards, the proposed method is particularly suited for near real-time applications that require a solution within a strict time frame, or when the problem size exceeds the hardware limitations of dedicated devices, such as current quantum computers.

Figures (8)

• Figure 1: Example of graph partitioning. The original graph, depicted on the top with gray nodes, is partitioned into two subgraphs, indicated by the different colors on the bottom. The lower figure shows how the local field $d_0^{(1)}$ accounts for the external interactions acting on node $0$. The definition of Eq. \ref{['eq:local_weights']} preserves the local structure of the problem by evenly balancing the impact of internal and external edges.
• Figure 2: Schematic representation of the SPLIT framework for quadratic programs (QP). The flow diagram outlines the key stages, starting with the graph partitioning step, which determines the variables $X_k$ to be assigned to each subproblem, followed by initialization, iterative updates, and the final convergence check. Each step is designed to balance computational efficiency with solution accuracy.
• Figure 3: Graph with $N=90$ nodes partitioned by spectral clustering. For this graph, the SPLIT algorithm identifies the exact solution of the MaxCut problem. The exact solution takes TTS$_{\rm CPLEX-Exact} = 258 \; {\rm s}$, while SPLIT takes TTS$_{\rm SPLIT} = 3.3 \; {\rm s}$, reaching the exact solution after $N_{\rm iter} = 5$.
• Figure 4: Upper panel: Boxplot showing the distribution of TTSs for SPLIT (blue boxes) and CPLEX Exact (orange boxes), on a semilog scale. Results for the MaxCut problem are displayed as a function of the number of graph nodes $N$ (30 instances for each problem size). The boxes represent the interquartile range. The horizontal lines within the boxes denote the median values. The whiskers span $1.5$ times the box range. Outliers located outside the range of the whiskers are displayed as empty circles. Lower panel: Log-log scatterplot of the speed-up of SPLIT as a function of the CPLEX Exact TTS for different instance sizes, highlighted by different colors.
• Figure 5: Approximation ratio \ref{['eq:alpha']} of the MaxCut solution with increasing $N$. The crosses represent the values of the approximation ratio achieved on the various instances. The average value of $\alpha$ for each graph size $N$ is shown by the dots connected by full lines, while the minimum approximation ratio observed among the different instances is indicated by the dashed lines. The blue markers indicate results obtained with SPLIT, while the red ones are CPLEX Approx results.