Recent studies suggest that gradient-based methods applied to relaxed box-constrained Quadratic Unconstrained Binary Optimization (QUBO) formulations can outperform classical heuristics in some large-scale regimes, often relying on heavy parallelization. However, these methods still underperform heuristics in other settings. In this work, we clarify this apparent discrepancy through a detailed analysis of the relaxed non-convex QUBO local maxima for both the Maximum Independent Set (MIS) and Maximum Cut (MaxCut) problems, and by introducing a new quadratic objective for MaxCut. Motivated by this analysis, we propose a mutation-based differentiable global reset algorithm, combined with local search to escape local maxima. We term our approach mQO, standing for mutation-based Quadratic combinatorial Optimization. The proposed strategy dramatically improves the performance of gradient-based solvers without heavy reliance on GPU parallelized initializations, indicating that stalling, rather than model capacity or compute, is the dominant bottleneck. As a result, on large-scale graphs, mQO achieves superior performance against state-of-the-art heuristics, commercial integer programming solvers, and recent GPU methods.

翻译：近期研究表明，应用于松弛箱约束二次无约束二元优化（QUBO）问题的梯度方法，在某些大规模场景下能超越经典启发式算法，且常依赖高度并行化。然而，这些方法在其他设定中表现仍逊于启发式算法。本文通过详细分析最大独立集（MIS）和最大割（MaxCut）问题中松弛非凸QUBO的局部极大值，并针对MaxCut引入新的二次目标函数，澄清了这一看似矛盾的现象。基于此分析，我们提出一种基于突变的可微全局重置算法，结合局部搜索以逃离局部极大值。我们将该方法命名为mQO（突变基二次组合优化）。所提策略显著提升了梯度求解器的性能，且无需过度依赖GPU并行的初始化过程，表明停滞而非模型容量或计算能力是主导瓶颈。因此，在大规模图上，mQO相较于最先进的启发式算法、商业整数规划求解器及近期GPU方法均取得了更优性能。