I blogged about the Douglas-Rachford method before here and here. It’s a method to solve monotone inclusions in the form

with monotone multivalued operators from a Hilbert space into itself. Using the resolvent and the reflector , the Douglas-Rachford iteration is concisely written as

The convergence of the method has been clarified is a number of papers, see, e.g.

Lions, Pierre-Louis, and Bertrand Mercier. “Splitting algorithms for the sum of two nonlinear operators.” SIAM Journal on Numerical Analysis 16.6 (1979): 964-979.

for the first treatment in the context of monotone operators and

Svaiter, Benar Fux. “On weak convergence of the Douglas–Rachford method.” SIAM Journal on Control and Optimization 49.1 (2011): 280-287.

for a recent very general convergence result.

Since is monotone if is monotone and , we can introduce a stepsize for the Douglas-Rachford iteration

It turns out, that this stepsize matters a lot in practice; too small and too large stepsizes lead to slow convergence. It is a kind of folk wisdom, that there is “sweet spot” for the stepsize. In a recent preprint Quoc Tran-Dinh and I investigated this sweet spot in the simple case of linear operators and and this tweet has a visualization.

A few days ago Walaa Moursi and Lieven Vandenberghe published the preprint “Douglas-Rachford splitting for a Lipschitz continuous and a strongly monotone operator” and derived some linear convergence rates in the special case they mention in the title. One result (Theorem 4.3) goes as follows: If is monotone and Lipschitz continuous with constant and is maximally monotone and -strongly monotone, than the Douglas-Rachford iterates converge strongly to a solution with a linear rate

This is a surprisingly complicated expression, but there is a nice thing about it: It allows to optimize for the stepsize! The rate depends on the stepsize as

and the two plots of this function below show the sweet spot clearly.

If one knows the Lipschitz constant of and the constant of strong monotonicity of , one can minimize to get on optimal stepsize (in the sense that the guaranteed contraction factor is as small as possible). As Moursi and Vandenberghe explain in their Remark 5.4, this optimization involves finding the root of a polynomial of degree 5, so it is possible but cumbersome.

Now I wonder if there is any hope to show that the adaptive stepsize Quoc and I proposed here (which basically amounts to in the case of single valued – note that the role of and is swapped in our paper) is able to find the sweet spot (experimentally it does).

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The fundamental theorem of calculus relates the derivative of a function to the function itself via an integral. A little bit more precise, it says that one can recover a function from its derivative up to an additive constant (on a simply connected domain).

In one space dimension, one can fix some in the domain (which has to be an interval) and then set

Then with .

Actually, a similar claim is true in higher space dimensions: If is defined on a simply connected domain in we can recover from its gradient up to an additive constant as follows: Select some and set

for any path from to . Then it holds under suitable conditions that

And now for something completely different: Convex functions and subgradients.

A function on is convex if for every there exists a *subgradient* such that for all one has the *subgradient inequality*

Writing this down for and with interchanged roles (and as corresponding subgradient to ), we see that

In other words: For a convex function it holds that the subgradient is a (multivalued) *monotone mapping*. Recall that a multivalued map is monotone, if for every and it holds that . It is not hard to see that not every monotone map is actually a subgradient of a convex function (not even, if we go to “maximally monotone maps”, a notion that we sweep under the rug in this post). A simple counterexample is a (singlevalued) linear monotone map represented by

(which can not be a subgradient of some , since it is not symmetric).

Another hint that monotonicity of a map does not imply that the map is a subgradient is that subgradients have some stronger properties than monotone maps. Let us write down the subgradient inequalities for a number of points :

If we sum all these up, we obtain

This property is called *-cyclical monotonicity*. In these terms we can say that a subgradient of a convex function is cyclical monotone, which means that it is -cyclically monotone for every integer .

By a remarkable result by Rockafellar, the converse is also true:

**Theorem 1 (Rockafellar, 1966)** * Let by a cyclically monotone map. Then there exists a convex function such that . *

Even more remarkable, the proof is somehow “just an application of the fundamental theorem of calculus” where we recover a function by its subgradient (up to an additive constant that depends on the basepoint).

*Proof:* we aim to “reconstruct” from . The trick is to choose some base point and corresponding and set

where the supremum is taken over all integers and all pairs . As a supremum of affine functions is clearly convex (even lower semicontinuous) and since is cyclically monotone (this shows that is *proper*, i.e. not equal to everywhere). Finally, for we have

with the supremum taken over all integers and all pairs . The right hand side is equal to and this shows that is indeed convex.

Where did we use the fundamental theorem of calculus? Let us have another look at equation~(0). Just for simplicity, let us denote . Now consider a path from to and points with . Then the term inside the supremum of~(0) equals

This is Riemannian sum for an integral of the form . By monotonicity of , we increase this sum, if we add another point (e.g. , and hence, the supremum does converge to the integral, i.e.~(0) is equal to

where is any path from to .

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