A penalized method for multivariate concave least squares with application to productivity analysis

TL;DR

This paper introduces a penalized quadratic programming approach for multivariate concave least squares estimation, significantly reducing computational time and enabling efficient shape-restricted regression analysis.

Contribution

It proposes a novel penalized reformulation of the multivariate concave least squares estimator as a simpler quadratic program with non-negativity constraints, improving computational efficiency.

Findings

Reformulated estimator is solved faster than the original.

The method applies to monotonic-concave/convex least squares.

Empirical results demonstrate improved computational performance.

Abstract

We propose a penalized method for the least squares estimator of a multivariate concave regression function. This estimator is formulated as a quadratic programming (QP) problem with $O(n^2)$ constraints, where n is the number of observations. Computing such an estimator is a very time-consuming task, and the computational burden rises dramatically as the number of observations increases. By introducing a quadratic penalty function, we reformulate the concave least squares estimator as a QP with only non-negativity constraints. This reformulation can be adapted for estimating variants of shape restricted least squares, i.e. the monotonic-concave/convex least squares. The experimental results and an empirical study show that the reformulated problem and its dual are solved significantly faster than the original problem. The Matlab and R codes for implementing the penalized problems are provided in the paper.