A Mixed Prinmal-Dual Bases Algorithm for Regression under Inequality Constraints. Application to Concave regression
提出一种有限步内求解不等式约束下最小二乘估计的算法，通过固定直线连接初始点与数据点，利用混合原始-对偶基快速得到投影，适用于凹回归等约束问题。
In some statistical non-parametric models the mean of the random variable y has to satisfy specific constraints. We consider the case where the set defined by the constraints is a closed polyhedral cone K in Rk. For example, when the mean is required to be concave in x, the set of acceptable means is a closed convex cone defined by k-2 linear inequalities in Rk. The least squares estimate of the mean is then the projection of the data point y on the cone K. In this paper, we present an algorithm to find the least square estimate of the mean in a finite number of steps. Other algorithms to solve this problem have been given before. The successive approximations in such algorithms are usually points on the faces of K. The solution here is reached by following a fixed line joining an abitrary but suitably chosen initial point y0 to the data point y. The 1- dimensional subspace spanned by the generators of the cone K is divided into 2' regions which can be described as the set of points with non-negative coordinates in mixed primal-dual bases relative to the cone K. Any point y belongs to one and only one of these regions SJ with corresponding basis 3j. The projection of y on K is immediately obtained from the expression of y in Oj by dropping the dual component of y.