ReLU双变量在随机混合整数程序中切片生成中的规范化
Normalization of ReLU Dual for Cut Generation in Stochastic Mixed-Integer Programs
AI总结：
本文提出通过规范化ReLU双变量以生成更强切片的方法，改进随机混合整数规划中的切片生成效率与质量。
摘要：
我们研究了修正线性单元（ReLU）双变量，这是一种已有的双变量公式用于随机规划，它利用ReLU函数重新表述非anticipativity约束以生成紧密、非凸且混合整数可表示的切片。尽管这种双变量重新公式化在混合整数状态变量中保证收敛，但它允许多个最优解，这些解可以产生弱切片。为了解决这个问题，我们提出在扩展空间中对双变量进行规范化，以识别产生更强切片的解。我们证明所得到的规范化切片在原始状态空间中是紧密的且帕累托最优的。我们进一步将规范化与现有基于正则化的处理双变量退化的做法进行比较，并解释为什么规范化提供了关键优势。特别是，我们显示规范化可以恢复任何通过正则化获得的切片，而反之则不成立。计算实验表明，所提出的方法在更难的实例上通过一致产生更强的切片并减少求解时间而优于现有方法。
英文摘要：
We study the Rectified Linear Unit (ReLU) dual, an existing dual formulation for stochastic programs that reformulates non-anticipativity constraints using ReLU functions to generate tight, non-convex, and mixed-integer representable cuts. While this dual reformulation guarantees convergence with mixed-integer state variables, it admits multiple optimal solutions that can yield weak cuts. To address this issue, we propose normalizing the dual in the extended space to identify solutions that yield stronger cuts. We prove that the resulting normalized cuts are tight and Pareto-optimal in the original state space. We further compare normalization with existing regularization-based approaches for handling dual degeneracy and explain why normalization offers key advantages. In particular, we show that normalization can recover any cut obtained via regularization, whereas the converse does not hold. Computational experiments demonstrate that the proposed approach outperforms existing methods by consistently yielding stronger cuts and reducing solution times on harder instances.
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