Could someone provide an example of a function that is monotonic increasing but not cyclically monotonic increasing?
This question comes from reading the statement on page 2 just before Prop 3 here
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1 Answer
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Consider an $N\times N$ matrix $A$ and the function $f(x)=Ax$. For $f$ to be cyclically monotone, you need $f$ to be the gradient of a convex function, and so $A$ must be symmetric and positive semi-definite. For $f$ to be monotone, it is enough that $A+A^T$ be positive semi-definite.
