An augmented Lagrangian method with constraint generation for shape-constrained convex regression problems

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 unified framework for computing the least squares estimator of a multivariate shape-constrained convex regression function in 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, n(n − 1) linear inequality constraints and n possibly non-polyhedral inequality constraints, where n is the number of data points. To efficiently solve the generally very large-scale convex QP, we design a proximal augmented Lagrangian method (pALM) whose subproblems are solved by the semismooth Newton method (SSN). To further accelerate the computation when n is huge, we design a practical implementation of the constraint generation method such that each reduced problem is efficiently solved by our proposed pALM. Comprehensive numerical experiments, including those in the pricing of basket options and estimation of production functions in economics, demonstrate that our proposed pALM outperforms the state-of-the-art algorithms, and the proposed acceleration technique further shortens the computation time by a large margin.

Forward-looking assessment of the GHG abatement cost: Application to China

“…Recent work has shown several promising algorithms from the perspective of computational burden (see, e.g., Lee et al, 2013;Balázs et al,2015;Mazumder et al, 2019;Bertsimas & Mundru, 2021;Lin et al 2020). It is worth highlighting two of them here, namely, the CNLS-G algorithm proposed by Lee et al (2013) and the cutting-plane algorithm proposed by Balázs et al (2015) and extended by Bertsimas & Mundru (2021).…”