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 ${x_{0}}$ in the domain (which has to be an interval) and then set $\displaystyle F(x) = \int_{x_{0}}^{x}f'(t) dt.$

Then ${F(x) = f(x) + c}$ with ${c= - f(x_{0})}$.

Actually, a similar claim is true in higher space dimensions: If ${f}$ is defined on a simply connected domain in ${{\mathbb R}^{d}}$ we can recover ${f}$ from its gradient up to an additive constant as follows: Select some ${x_{0}}$ and set $\displaystyle F(x) = \int_{\gamma}\nabla f(x)\cdot dx \ \ \ \ \ (1)$

for any path ${\gamma}$ from ${x_{0}}$ to ${x}$. Then it holds under suitable conditions that $\displaystyle F(x) = f(x) - f(x_{0}).$

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

A function ${f}$ on ${{\mathbb R}^{n}}$ is convex if for every ${x}$ there exists a subgradient ${x^{*}\in{\mathbb R}^{n}}$ such that for all ${y}$ one has the subgradient inequality $\displaystyle f(y) \geq f(x) + \langle x^{*}, y-x\rangle.$

Writing this down for ${x}$ and ${y}$ with interchanged roles (and ${y^{*}}$ as corresponding subgradient to ${y}$), we see that $\displaystyle \langle x-y,x^{*}-y^{*}\rangle \geq 0.$

In other words: For a convex function ${f}$ it holds that the subgradient ${\partial f}$ is a (multivalued) monotone mapping. Recall that a multivalued map ${A}$ is monotone, if for every ${y \in A(x)}$ and ${y^{*}\in A(y)}$ it holds that ${\langle x-y,x^{*}-y^{*}\rangle \geq 0}$. 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 $\displaystyle \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}$

(which can not be a subgradient of some ${f}$, 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 ${x_{0},\dots x_{n}}$: $\displaystyle \begin{array}{rcl} f(x_{1}) & \geq & f(x_{0}) + \langle x_{0}^{*},x_{1}-x_{0}\rangle\\ f(x_{2}) & \geq & f(x_{1}) + \langle x_{1}^{*},x_{2}-x_{1}\rangle\\ \vdots & & \vdots\\ f(x_{n}) & \geq & f(x_{n-1}) + \langle x_{n-1}^{*},x_{n}-x_{n-1}\rangle\\ f(x_{0}) & \geq & f(x_{n}) + \langle x_{n}^{*},x_{0}-x_{n}\rangle. \end{array}$

If we sum all these up, we obtain $\displaystyle 0 \geq \langle x_{0}^{*},x_{1}-x_{0}\rangle + \langle x_{1}^{*},x_{2}-x_{1}\rangle + \cdots + \langle x_{n-1}^{*},x_{n}-x_{n-1}\rangle + \langle x_{n}^{*},x_{0}-x_{n}\rangle.$

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

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

Theorem 1 (Rockafellar, 1966) Let ${A}$ by a cyclically monotone map. Then there exists a convex function ${f}$ such that ${A = \partial f}$.

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” ${f}$ from ${A = \partial f}$. The trick is to choose some base point ${x_{0}}$ and corresponding ${x_{0}^{*}\in A(x_{0})}$ and set $\displaystyle \begin{array}{rcl} f(x) &=& \sup\left\{ \langle x_{0}^{*}, x_{1}-x_{0}\rangle + \langle x_{1}^{*},x_{2}-x_{1}\rangle + \cdots + \langle x_{n-1}^{*},x_{n}-x_{n-1}\rangle\right.\\&& \qquad\left. + \langle x_{n}^{*},x-x_{n}\rangle\right\}\qquad\qquad (0)\end{array}$

where the supremum is taken over all integers ${m}$ and all pairs ${x_{i}^{*}\in A(x_{i})}$ ${i=1,\dots, m}$. As a supremum of affine functions ${f}$ is clearly convex (even lower semicontinuous) and ${f(x_{0}) = 0}$ since ${A}$ is cyclically monotone (this shows that ${f}$ is proper, i.e. not equal to ${\infty}$ everywhere). Finally, for ${\bar x^{*}\in A(\bar x)}$ we have $\displaystyle \begin{array}{rcl} f(x) &\geq& \sup\left\{ \langle x_{0}^{*}, x_{1}-x_{0}\rangle + \langle x_{1}^{*},x_{2}-x_{1}\rangle + \cdots + \langle x_{n-1}^{*},x_{n}-x_{n-1}\rangle\right.\\ & & \qquad \left.+ \langle x_{n}^{*},\bar x-x_{n}\rangle + \langle \bar x^{*}, x-\bar x\rangle\right\} \end{array}$

with the supremum taken over all integers ${m}$ and all pairs ${x_{i}^{*}\in A(x_{i})}$ ${i=1,\dots, m}$. The right hand side is equal to ${f(x) + \langle \bar x^{*},x-\bar x\rangle}$ and this shows that ${f}$ is indeed convex. $\Box$

Where did we use the fundamental theorem of calculus? Let us have another look at equation~(0). Just for simplicity, let us denote ${x_{i}^{*} =\nabla f(x_{i})}$. Now consider a path ${\gamma}$ from ${x_{0}}$ to ${x}$ and points ${0=t_{0}< t_{1}<\cdots < t_{n}< t_{n+1} = 1}$ with ${\gamma(t_{i}) = x_{i}}$. Then the term inside the supremum of~(0) equals $\displaystyle \langle \nabla f(\gamma(t_{0})),\gamma(t_{1})-\gamma(t_{0})\rangle + \dots + \langle \nabla f(\gamma(t_{n})),\gamma(t_{n+1})-\gamma(t_{n})\rangle.$

This is Riemannian sum for an integral of the form ${\int_{\gamma}\nabla f(x)\cdot dx}$. By monotonicity of $\nabla {f}$, we increase this sum, if we add another point ${\bar t}$ (e.g. ${t_{i}<\bar t, and hence, the supremum does converge to the integral, i.e.~(0) is equal to $\displaystyle f(x) = \int_{\gamma}\nabla f(x)\cdot dx$

where ${\gamma}$ is any path from ${x_{0}}$ to ${x}$.