# Is there an intuition for cyclic monotonicity?

Cyclic monotonicity says that if we have a correspondence, $x(w)$, the $x$ is cyclically monotone in $w$ if for a finite sequence $w_1,\cdots w_k$ and $x^*(w_i)\in x(w_i)$, we have $\sum_{i=1}^k (w_i-w_{i+1})\cdot x^*(w_i) \leq 0$ (I guess this is technically the definition for cyclically monotone increasing).

I am wondering if there is some intuition (or clear) explanation of what this says.

To me, it seems to say that when $w_{i+1} >w_i$ then $x_i > x_{i+1}$, in some sort of "general" sense.

R.T. Rockefellar was the convex analyst who showed that a (multivalued) linear operator is the subdifferential of a convex function iff the operator is cyclically monotone. To quote one of his papers on that topic:

The cyclic monotonicity condition can be viewed heuristically as a discrete substitute for two classical conditions: that a smooth convex function has a positive semi-definite second differential, and that all circuit integrals of an integrable vector field must vanish.

• Can you give some intuition about this property vs the general statement of maximal monotonicity? It's clear that maximal monotone implies maximal cyclically monotone. However, when I read papers, they almost always consdier maximal monotone operators and not non maximal cyclically monotone operators - why? Feb 21, 2023 at 15:10

Here's my very intuitive, hand-wavey approach - apologies for using the opposite sign convention, following Wikipedia, which makes more sense to me.

It helps to start thinking about this in the $$k=2$$ case in $$\mathbb R$$ first. There, you have that $$x$$ is cyclically monotone iff $$w_1 x^*(w_1)+w_2 x^*(w_2)\ge w_1 x^*(w_2)+w_2 x^*(w_1).$$ If $$x$$ is actually monotone increasing, then the LHS is where we "multiply the bigger $$w_i$$ by the bigger $$x^*(w_i)$$." The same holds for larger $$k$$, e.g. if $$k=3$$ then $$w_1 x^*(w_1)+w_2x^*(w_2)+w_3x^*(w_3)\ge w_1x^*(w_2)+w_2x^*(w_3)+w_3x^*(w_1)$$ whenever $$x$$ is monotone increasing, because we "multiply the biggest $$w_i$$ by the biggest $$x^*(w_i)$$, the second-biggest $$w_i$$ by the second-biggest $$x^*(w_i)$$, and so on."

In the $$\mathbb{R}^n$$ case, we compare in multiple directions. Here I find starting with $$k=3$$ to be more helpful intuitively. Now, cyclic monotonicity requires $$w_1x^*(w_1)+w_2x^*(w_2)+w_3x^*(w_3)\ge w_1x^*(w_2)+w_2x^*(w_3)+w_3x^*(w_1).$$ I think of this as being "$$x^*(w_1)$$ is the vector which points the most in the $$w_1$$ direction, as opposed to the $$w_2$$ and $$w_3$$ directions". That is:

• If all the vectors are off in the same direction from $$0$$, then the largest $$w_i$$ is multiplied by the largest $$x^*(w_i)$$.
• If the vectors all point in different directions, then the $$x^*(w_i)$$ vector points the most in the actual $$w_i$$ direction. In theory, I think this wouldn't need to be strict, i.e. a priori there's nothing which says $$x^*(w_2)$$ couldn't point slightly more in the $$w_1$$ direction than $$x^*(w_1)$$ does, as long as (say) $$w_3x^*(w_3)$$ was large enough. In fact, I suspect that the subdifferential-of-convex-function result above may mean this doesnt.