[R] OT -- isotonic regression subject to bound constraints.

Rolf Turner r.turner at auckland.ac.nz
Mon Dec 11 19:15:48 CET 2017

Well, I could argue that it's not *completely* OT since my question is 
motivated by an enquiry that I received in respect of a CRAN package 
"Iso" that I wrote and maintain.

The question is this:  Given observations y_1, ..., y_n, what is the 
solution to the problem:

   minimise \sum_{i=1}^n (y_i - y_i^*)^2

with respect to y_1^*, ..., y_n^* subject to the "isotonic" constraint
y_1^* <= y_2^* <= ... <= y_n^* and the *additional8 bound constraint
a <= y_1^* and y_n^* <= b, where a and b are given constants?

I have googled around a bit (unsuccessfully) and have asked this 
question on crossvalidated a couple of days ago, with no response whatever.

So I thought that I might try the super-knowledgeable R community, in 
the hope that someone out there might be able to tell me something useful.

Note that the question can be expressed as finding the projection of the 
point (y_1, ..., y_n) onto the intersection of the isotonic cone and
the hypercube [a,b]^n.

At first I thought that protecting onto the isotonic cone and then 
projection that result onto the hypercube might work, but I am now 
pretty sure that is hopelessly naive.

Any hints?  Ta.


Rolf Turner

Technical Editor ANZJS
Department of Statistics
University of Auckland
Phone: +64-9-373-7599 ext. 88276

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