Efficient algorithms for multivariate shape-constrained convex regression problems

• URL: http://arxiv.org/abs/2002.11410v1
• Date: Wed, 26 Feb 2020 11:18:43 GMT
• Title: Efficient algorithms for multivariate shape-constrained convex regression problems
• Authors: Meixia Lin, Defeng Sun, Kim-Chuan Toh
• Abstract summary: We prove that the least squares estimator is computable via solving a constrained convex programming (QP) problem with $(n+1)d$ variables and at least $n(n-1)$ linear inequality constraints. 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 (t
• Score: 9.281671380673306
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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