*Accepted Paper*

**Inserted:** 1 sep 2009

**Last Updated:** 30 oct 2009

**Journal:** Journal of Nonparametric Statistics

**Year:** 2009

**Notes:**

This is not a true calculus of variations paper. It's an application of convex optimization to statistics. But, curiously enough, the main issue which is addressed (L^{2} projection on increasing functions) is used in some optimal transport papers as well.

**Abstract:**

To project (in the $L^2$ metric) a single function $f$ on the set of increasing functions one takes the primitive of $f$, then its convex hull, and then takes the derivative. Here the problem is more complicated: project a pair $(f,g)$ on the set of pairs of increasing functions, the former smaller than the latter. It is a regression problem with applications in statistics. The solution is not explicit but we solve it numerically through a projected subgradient algorithm from convex optimization.

**Keywords:**
regression, monotone curves, subgradient descent

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