We propose a new class of convex approximations for two-stage mixed-integer recourse models, the so-called generalized alpha-approximations. The advantage of these convex approximations over existing ones is that they are more suitable for efficient computations. Indeed, we construct a loose Benders decomposition algorithm that solves large problem instances in reasonable time. To guarantee the performance of the resulting solution, we derive corresponding error bounds that depend on the total variations of the probability density functions of the random variables in the model. The error bounds converge to zero if these total variations converge to zero. We empirically assess our solution method on several test instances, including the SIZES and SSLP instances from SIPLIB. We show that our method finds near-optimal solutions if the variability of the random parameters in the model is large. Moreover, our method outperforms existing methods in terms of computation time, especially for large problem instances.

中文翻译：

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一种近似两阶段混合整数资源模型的松散Benders分解算法

我们为两阶段混合整数资源模型提出了一类新的凸近似，即所谓的广义 alpha 近似。这些凸近似比现有的优势在于它们更适合高效计算。事实上，我们构建了一个松散的 Benders 分解算法，可以在合理的时间内解决大型问题实例。为了保证结果解决方案的性能，我们推导出相应的误差界限，该界限取决于模型中随机变量的概率密度函数的总变化。如果这些总变化收敛到零，则误差界限收敛到零。我们在几个测试实例上凭经验评估我们的解决方案方法，包括来自 SIPLIB 的 SIZES 和 SSLP 实例。我们表明，如果模型中随机参数的可变性很大，我们的方法可以找到接近最优的解决方案。此外，我们的方法在计算时间方面优于现有方法，尤其是对于大型问题实例。