论文标题
非线性优化问题的经验分位数估计方法与机会限制
An Empirical Quantile Estimation Approach to Nonlinear Optimization Problems with Chance Constraints
论文作者
论文摘要
我们研究了一种解决机会约束的非线性优化问题的经验分位数估计方法。我们的方法是基于对机会约束的重新重新制定，作为等效的分位数约束，以在梯度上提供更强的信号。在这种方法中，分位数函数的值是从从随机参数中得出的样品进行经验估算的，并且通过在分位数函数值估计的顶部通过有限差近似值来估算分位数函数的梯度。我们在增强的拉格朗日方法的框架内建立了这种方法的收敛理论，该方法用于解决一般的非线性约束优化问题。收敛分析的基础是经验分位数过程的浓度特性，并且根据分位数函数是否可区分分析分析。与文献中使用的采样和平滑方法相反，本文开发的方法不涉及任何平滑函数，因此更容易实现分位数功能梯度近似，并且精确控制参数可以调节。我们证明了这种方法的有效性，并将其与平滑方法进行比较，以进行分位数梯度估计。数值调查表明，这两种方法在某些问题实例中具有竞争力。
We investigate an empirical quantile estimation approach to solve chance-constrained nonlinear optimization problems. Our approach is based on the reformulation of the chance constraint as an equivalent quantile constraint to provide stronger signals on the gradient. In this approach, the value of the quantile function is estimated empirically from samples drawn from the random parameters, and the gradient of the quantile function is estimated via a finite-difference approximation on top of the quantile-function-value estimation. We establish a convergence theory of this approach within the framework of an augmented Lagrangian method for solving general nonlinear constrained optimization problems. The foundation of the convergence analysis is a concentration property of the empirical quantile process, and the analysis is divided based on whether or not the quantile function is differentiable. In contrast to the sampling-and-smoothing approach used in the literature, the method developed in this paper does not involve any smoothing function and hence the quantile-function gradient approximation is easier to implement and there are less accuracy-control parameters to tune. We demonstrate the effectiveness of this approach and compare it with a smoothing method for the quantile-gradient estimation. Numerical investigation shows that the two approaches are competitive for certain problem instances.
