Deriving robust counterparts of nonlinear uncertain inequalities
Deriving robust counterparts of nonlinear uncertain inequalities
复制标题
导出非线性不确定不等式的鲁棒对应项
DOI：
10.1007/s10107-014-0750-8
复制
发表时间：
2014
影响因子：
2.7
通讯作者：
J. Vial
中科院分区：
文献类型：
--
作者：
A. Ben;Dick den Hertog;J. Vial
本文提供了一种系统的方法来构造在不确定参数中为凹的非线性不确定不等式的鲁棒对应不等式。我们使用凸分析（支持函数，共轭函数，Fenchel对偶）和圆锥对偶，以便将强大的对应转换成一个明确的和计算上易于处理的约束。事实证明，要做到这一点，必须计算不确定性集的支持函数和非线性约束函数的凹共轭。方便的是，这两个计算完全独立。这种方法有几个优点。首先，它提供了一个简单的结构化的方法来构造鲁棒的线性和非线性不等式。第二，它表明，对于新的类的不确定性区域和新的类的非线性优化问题，可以推导出听话的同行。我们还研究了不等式在不确定参数下是非凹的情况。
In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.