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

Shape-constrained convex regression problem deals with fitting a function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides unified framework for computing least squares estimator of multivariate shape-constrained in $${\mathbb {R}}^d$$ . We prove that is computable via solving an essentially constrained quadratic programming (QP) $$(d+1)n$$ variables, $$n(n-1)$$ linear inequality n possibly non-polyhedral constraints, number data points. To efficiently solve generally very large-scale QP, we design proximal augmented Lagrangian method (proxALM) whose subproblems solved by semismooth Newton method. further accelerate computation when huge, practical implementation constraint generation each reduced our proposed proxALM. Comprehensive numerical experiments, including those pricing basket options estimation production functions economics, demonstrate proxALM outperforms state-of-the-art algorithms, acceleration technique shortens time large margin.