  This paper studies the convex isotonic regression with generalized order
restrictions induced by a directed tree. The proposed model covers various
intriguing optimization problems with shape or order restrictions, including
the generalized nearly isotonic optimization and the total variation on a tree.
Inspired by the success of the pool-adjacent-violator algorithm and its
active-set interpretation, we propose an active-set based recursive approach
for solving the underlying model. Unlike the brute-force approach that
traverses an exponential number of possible active-set combinations, our
algorithm has a polynomial time computational complexity under mild
assumptions.
