Subgradient

Vector in $R^{n-1}$ at which subgradient should be computed.

Vector in $R^n$, measurements of upper function.

Vector in $R^n$, weights for upper function.

Vector in $R^n$, measurements of lower function.

Vector in $R^n$, weights for lower function.

Matrix in $R^{n \times n}$ that contains the quantities $G_{s, i}$ defined
  in Balabdaoui et al. (2009).

This function computes a subgradient of the function $\Psi$.

regression

nonparametric

We consider the problem of estimating two isotonic regression curves g1* and g2* under the constraint that they are ordered, i.e. g1* <= g2*. Given two sets of n data points y_1, ..., y_n and z_1, ..., z_n that are observed at (the same) deterministic design points x_1, ..., x_n, the estimates are obtained by minimizing the Least Squares criterion L(a, b) = sum_{i=1}^n (y_i - a_i)^2 w1(x_i) + sum_{i=1}^n (z_i - b_i)^2 w2(x_i) over the class of pairs of vectors (a, b) such that a and b are isotonic and a_i <= b_i for all i = 1, ..., n. We offer two different approaches to compute the estimates: a projected subgradient algorithm where the projection is calculated using a PAVA as well as Dykstra's cyclical projection algorithm.

Subgradient: Computes a subgradient for the projected subgradient algorithm

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