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This paper is concerned with regression under a “sum” of partial order constraints. Examples include locally monotonic, piecewise monotonic, runlength constrained, and unimodal and oligomodal regression. These are of interest not only in nonlinear filtering but also in density estimation and chromatographic analysis. It is shown that under a least absolute error criterion, these problems can be transformed into appropriate finite problems, which can then be efficiently solved via dynamic programming techniques. Although the result does not carry over to least squares regression, hybrid programming algorithms can be developed to solve least squares counterparts of certain problems in the class