Ising machines (IM) are physics-inspired alternatives to von Neumann architectures for solving hard optimization tasks. By mapping binary variables to coupled Ising spins, IMs can naturally solve unconstrained combinatorial optimization problems such as finding maximum cuts in graphs. However, despite their importance in practical applications, constrained problems remain challenging to solve for IMs that require large quadratic energy penalties to ensure the correspondence between energy ground states and constrained optimal solutions. To relax this requirement, we propose a self-adaptive IM that iteratively shapes its energy landscape using a Lagrange relaxation of constraints and avoids prior tuning of penalties. Using a probabilistic-bit (p-bit) IM emulated in software, we benchmark our algorithm with multidimensional knapsack problems (MKP) and quadratic knapsack problems (QKP), the latter being an Ising problem with linear constraints. For QKP with 300 variables, the proposed algorithm finds better solutions than state-of-the-art IMs such as Fujitsu's Digital Annealer and requires 7,500x fewer samples. Our results show that adapting the energy landscape during the search can speed up IMs for constrained optimization.

翻译：伊辛机（IM）是受物理学启发的冯·诺依曼架构替代方案，用于解决复杂优化任务。通过将二元变量映射到耦合的伊辛自旋，IM能够自然地解决无约束组合优化问题，例如寻找图中的最大割。然而，尽管在实际应用中具有重要意义，约束问题对于IM而言仍然具有挑战性，因为需要较大的二次能量惩罚来确保能量基态与约束最优解之间的对应关系。为了放宽这一要求，我们提出了一种自适应IM，它利用约束的拉格朗日松弛迭代地塑造其能量景观，并避免预先调整惩罚参数。使用在软件中模拟的概率比特（p-bit）IM，我们通过多维背包问题（MKP）和二次背包问题（QKP）对算法进行基准测试，后者是具有线性约束的伊辛问题。对于包含300个变量的QKP，所提出的算法找到了比最先进的IM（如富士通数字退火器）更好的解，并且所需的样本量减少了7,500倍。我们的结果表明，在搜索过程中调整能量景观可以加速IM在约束优化中的应用。