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Shape-constrained convex regression problem deals with fitting a convexfunction to the observed data, where additional constraints are imposed, suchas component-wise monotonicity and uniform Lipschitz continuity. This paperprovides a comprehensive mechanism for computing the least squares estimator ofa multivariate shape-constrained convex regression function in $\mathbb{R}^d$.We prove that the least squares estimator is computable via solving aconstrained convex quadratic programming (QP) problem with $(n+1)d$ variablesand at least $n(n-1)$ linear inequality constraints, where $n$ is the number ofdata points. For solving the generally very large-scale convex QP, we designtwo efficient algorithms, one is the symmetric Gauss-Seidel based alternatingdirection method of multipliers ({\tt sGS-ADMM}), and the other is the proximalaugmented Lagrangian method ({\tt pALM}) with the subproblems solved by thesemismooth Newton method ({\tt SSN}). Comprehensive numerical experiments,including those in the pricing of basket options and estimation of productionfunctions in economics, demonstrate that both of our proposed algorithmsoutperform the state-of-the-art algorithm. The {\tt pALM} is more efficientthan the {\tt sGS-ADMM} but the latter has the advantage of being simpler toimplement.

Efficient algorithms for multivariate shape-constrained convex regression problems