论文标题
多元形状约束凸回归问题的有效算法
Efficient algorithms for multivariate shape-constrained convex regression problems
论文作者
论文摘要
形状受限的凸回归问题涉及将凸功能拟合到观察到的数据，在该数据中施加了其他约束，例如组件的单调性和统一的Lipschitz连续性。本文提供了一种综合机制，用于计算$ \ mathbb {r}^d $中多元形状约束凸回归函数的最小二乘估计器。我们证明，使用$（n+1）d $变量和至少$ n（n-1）$线性不平等约束，$ n $是$ n $是数据点的数量，我们可以通过求解受约束的凸二次编程（QP）问题来计算最小二乘估计器。为了求解一般非常大的凸QP，我们设计了两种有效的算法，一个是基于对称的高斯 - seidel乘数的交替方向方法（{\ tt sgs-admm}），另一个是近端增强lagmagian方法（{\ tt Palm}）的近端增强方法（{\ tt Palm}），{\ tt Palm} from finder from septoved fifd ryms { SSN}）。全面的数值实验，包括篮子期权定价和经济学生产功能的估计的实验，这表明，我们提出的两种算法的表现都优于最先进的算法。 {\ tt棕榈}比{\ tt sgs-admm}更有效，但是后者的优势是更简单地实现。
Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides a comprehensive mechanism for computing the least squares estimator of a multivariate shape-constrained convex regression function in $\mathbb{R}^d$. We prove that the least squares estimator is computable via solving a constrained convex quadratic programming (QP) problem with $(n+1)d$ variables and at least $n(n-1)$ linear inequality constraints, where $n$ is the number of data points. For solving the generally very large-scale convex QP, we design two efficient algorithms, one is the symmetric Gauss-Seidel based alternating direction method of multipliers ({\tt sGS-ADMM}), and the other is the proximal augmented Lagrangian method ({\tt pALM}) with the subproblems solved by the semismooth Newton method ({\tt SSN}). Comprehensive numerical experiments, including those in the pricing of basket options and estimation of production functions in economics, demonstrate that both of our proposed algorithms outperform the state-of-the-art algorithm. The {\tt pALM} is more efficient than the {\tt sGS-ADMM} but the latter has the advantage of being simpler to implement.
