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I would like to fit an isotonic regression (with respect to "ordinary" linear ordering) to observations y_1, ..., y_n subject to the additional constraints that a <= y_1^* and y_10^* <= b where a and b are given constants and the y_j^* are the fitted values.

In other words, find

 y_1^*, ..., y_n^*

which minimise

sum_{i=1}^n (y_i - y_i^*)^2 subject to the constraints

(1) y_1^* <= y_2^* <= ... <= y_n^*, and

(2) a <= y_1^, y_n^ <= b where a and b are given constants.

Is there a known solution to this problem?

Note that it could be phrased as finding the projection onto the intersection of the isotonic cone and the hypercube [a,b]^n.

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It turns out that the R package OrdMonReg, which is available on CRAN, handles this problem (and more).

The person who instigated my original question (via an email enquiry about my "Iso" package) found the OrdMonReg package and let me know about it.

The theoretical background is to be found in the paper:

"Least-squares estimation of two-ordered monotone regression curves" by Fadoua Balabdaouia, Kaspar Rufibachc and Filippo Santambrogioa, Journal of Nonparametric Statistics Vol. 22, No. 8, November 2010, pp. 1019–1037

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