Robust optimization has emerged as a powerful and efficient methodology for incorporating uncertain parameters into optimization models. In robust optimization, robust counterparts for uncertain constraints are created by imposing a known set of uncertain parameter realizations onto the new robust constraint. For constraints with all bounded parameters, the interval + ellipsoidal and interval + polyhedral uncertainty sets are well-established in robust optimization literature, while box, ellipsoidal, or polyhedral sets may be used for unbounded parameters. However, there has yet to be any counterparts proposed for constraints that simultaneously contain both bounded and unbounded parameters. This is crucial, as using the traditional box, ellipsoidal, or polyhedral sets with bounded parameters may impose impossible parameter realizations outside of their bounds, unnecessarily increasing the conservatism of results. In this work, robust counterparts for uncertain constraints with both bounded and unbounded uncertain parameters are derived: the generalized interval + box, generalized interval + ellipsoidal, and generalized interval + polyhedral counterparts. These counterparts reduce to the traditional box, ellipsoidal, and polyhedral counterparts if all parameters are unbounded, and reduce to the traditional interval + ellipsoidal and interval + polyhedral counterparts if all parameters are bounded. It is proven that established a priori probabilistic bounds remain valid for these counterparts. The importance of these developments is demonstrated with computational examples, showing the reduction of conservatism that is gained by appropriately limiting the possible realizations of the bounded parameters. The developments increase the scope and applicability of robust optimization as a tool for optimization under uncertainty.

稳健的优化已成为一种将不确定的参数合并到优化模型中的强大而有效的方法。在鲁棒优化中，通过将一组已知的不确定参数实现强加到新的鲁棒约束上来创建不确定约束的鲁棒对应项。对于具有所有有界参数的约束，区间+椭圆和区间+多面体不确定性集在稳健的优化文献中已得到很好的建立，而箱形，椭圆或多面体集可用于无界参数。但是，对于约束同时包含有界和无界参数的约束，尚未提出任何建议。这是至关重要的，因为使用传统的椭圆形盒子 或带有有界参数的多面体集可能会在其界限外强加不可能的参数实现，从而不必要地增加了结果的保守性。在这项工作中，导出了具有有界和无界不确定参数的不确定约束的鲁棒对应物：广义区间+框，广义区间+椭圆形以及广义区间+多面体对应物。如果所有参数都无界，则这些对应项简化为传统的矩形，椭圆形和多面体对应项，如果所有参数都为有界，则减小为传统的间隔+椭圆形和区间+多面体对应项。事实证明，建立 导出了具有有界和无界不确定参数的不确定约束的鲁棒对应物：广义区间+框，广义区间+椭圆形以及广义区间+多面体对应物。如果所有参数都无界，则这些对应项简化为传统的矩形，椭圆形和多面体对应项；如果所有参数都为有界，则减小为传统的间隔+椭圆形和区间+多面体对应项。事实证明，建立 导出了具有有界和无界不确定参数的不确定约束的鲁棒对应物：广义区间+框，广义区间+椭圆形以及广义区间+多面体对应物。如果所有参数都无界，则这些对应项简化为传统的矩形，椭圆形和多面体对应项，如果所有参数都为有界，则减小为传统的间隔+椭圆形和区间+多面体对应项。事实证明，建立 如果所有参数都有界，则减小到传统的区间+椭圆形和区间+多面体对应物。事实证明，建立 如果所有参数都有界，则减小到传统的区间+椭圆形和区间+多面体对应物。事实证明，建立先验概率界限对于这些对等物仍然有效。这些发展的重要性通过计算示例得到了证明，这些计算示例显示了通过适当地限制有界参数的可能实现而获得的保守性的降低。这些发展增加了鲁棒优化作为不确定性条件下优化工具的范围和适用性。