Efficient algorithms for multivariate shape-constrained convex regression problems
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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 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 ( sGS-ADMM), and the other is the proximal augmented Lagrangian method ( pALM) with the subproblems solved by the semismooth Newton method ( 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 pALM is more efficient than the sGS-ADMM but the latter has the advantage of being simpler to implement.
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