Chain rule of convex function

This post studies a specific chain rule of composition of convex functions. Specifically, we have the following theorem.

Theorem 1 For a continuously differentiable increasing function {\phi: \mathbb{R} \rightarrow \mathbb{R}}, a convex function {h: U \rightarrow \mathbb{R}} where {U\in \mathbb{R}^n} is a convex set and an {x\in U}, if {\phi'(h(x))>0} or {x\in \mbox{int}(U)}, then

\displaystyle \begin{aligned} \partial (\phi \circ h) (x) = \phi' (h(x)) \cdot[ \partial h (x)], \end{aligned} \ \ \ \ \ (1)

where {\partial } is the operator of taking subdifferentials of a function, i.e., {\partial h (x) = \{ g\mid h(x)\geq h(y) +\langle{g},{y-x}\rangle,\forall y\in U\}} for any {x\in U}, and {\mbox{int}(U)} is the interior of {U} with respect to the standard topology in {\mathbb{R}^n}.

A negative example. We note that if our condition fails, the inequality may not hold. For example, let {\phi(x) =1} for all {x\in \mathbb{R}} and {h(x) = 1 } defined on {[0,1]}. Then {0} is a point which is not in the interior of {[0,1],\phi'(0) = 0}, {\partial h(0) =(-\infty,0]}. However, in this case {\partial (\phi\circ h)(0)= (-\infty,0]} and {\phi' (h(0)) \cdot[ \partial h (0)] =0}. Thus, the equality fails.

It should be noted that if {U} is open and {h} is also differentiable, then the above reduces to the common chain rule of smooth functions.

Proof: We first prove that {\partial (\phi \circ h) (x) \supset \phi' (h(x)) \cdot[ \partial h (x)]}. We have for all {x\in U , g \in \partial h(x)},

\displaystyle \begin{aligned} \phi (h(y)) &\overset{(a)}{\geq} \phi(h(x)) + \phi' (h(x))(h(y)-h(x))\\ & \overset{(b)}{\geq} \phi (h(x)) + \phi '(h(x)) \langle g,y-x\rangle \\ & = \phi (h(x)) + \langle{\phi' (h(x))g},{y-x}\rangle \end{aligned} \ \ \ \ \ (2)

where {(a),(b)} are just the definition of subdifferential of {\phi} at {h(x)} and {h} at {x}. We also use the fact that {\phi(x)\geq 0} in the inequality {(b)} as {\phi} is increasing.

Now we prove the other direction. Without lost of generality, suppose {0\in U} such that {\partial (\phi \circ h)(0)} is not empty. Let {g\in \partial (\phi \circ h)(0)}, we wish to show that {g} is in the set { \phi' (h(0)) \cdot[ \partial h (0)]}. First according to the definition of subdifferential, we have

\displaystyle \begin{aligned} (\phi \circ h) (x)\geq (\phi\circ h)(0) + \langle{ g},x\rangle, \forall x \in U \end{aligned} \ \ \ \ \ (3)

This gives

\displaystyle \begin{aligned} (\phi \circ h) (\gamma x)\geq (\phi\circ h)(0) + \langle{ g} ,{\gamma x}\rangle, \forall x \in U, \gamma \in [0,1]. \end{aligned} \ \ \ \ \ (4)

Rearranging the above inequality gives

\displaystyle \begin{aligned} &\frac{(\phi \circ h) (\gamma x)- (\phi\circ h)(0)}{\gamma}\geq \langle{ g},{ x}\rangle \\ \implies & \phi'(s)\cdot \frac{h(\gamma x)-h(0)}{\gamma}\geq \langle{g},{x}\rangle \end{aligned} \ \ \ \ \ (5)

for some {s} between {h(\gamma x)} and {h(0)} by mean value theorem. Now, by letting {f(\gamma )=h(\gamma x)}, if {f} is right continuous at {0}, then by Lemma 1 in this post and {\phi} is continuously differentiable, we know {l(\gamma) =\frac{h(\gamma x)-h(0)}{\gamma}} is nondecreasing in {\gamma} and we have

\displaystyle \begin{aligned} \phi'(h(0)) (h(x)-h(0))\geq \phi'(h(0))\cdot \lim_{\gamma \downarrow 0} \frac{h(\gamma x)-h(0)}{\gamma}\geq \langle{g},{x}\rangle,\forall x\in U. \end{aligned} \ \ \ \ \ (6)

If {\phi' (h(0))>0}, then dividing both sides of the above inequality by {\phi'(h(0))} gives

\displaystyle \begin{aligned} h(x)-h(0)\geq \langle{\frac{g}{\phi'(h(0))}},{x}\rangle,\forall x\in U. \end{aligned} \ \ \ \ \ (7) 

This shows that {\frac{g}{\phi'(h(0))}} is indeed a member of {\partial h(x)} and thus {\partial (\phi \circ h) (x) \subset \phi' (h(x)) \cdot[ \partial h (x)]}. In this case, we only need to verify why {f(\gamma ) = h(\gamma x) } must be right continuous at {0}.

If {0\in \mbox{int}(U)}, then {h} is definitely continuous at {x} and so is {f} by standard result in convex analysis. If {\phi' (h(0))>0}, then we are done by inequality (7). If {\phi' (h(0))=0}, using the inequality (6), we have

\displaystyle \begin{aligned} 0 \geq \langle{g},{x}\rangle, \forall x\in U \end{aligned} \ \ \ \ \ (8)

Since {0 \in \mbox{int}(U)}, then {x} can take a small positive and negative multiple of {n} standard basis vectors in {\mathbb{R}^n} in the inequality (8). This shows {g =0} and it indeed belongs to the set {\phi' (h(0)) \cdot[ \partial h (x)] = \{0\}} as {\partial (h(0))\not=\emptyset} for {0\in \mbox{int}(U)} by standard convex analysis result.

Thus our task now is to argue why {f(\gamma ) = h(\gamma x)} is indeed right continuous at {0}. Using Lemma 4 in this post, we know the limit {\lim_{\gamma \downarrow 0 }f(\gamma) = f(0^+)} exists and {f(0^+)\leq f(0)}. Now if {f(0^+) = f(0) = h(0)}, then {f} is indeed right continuous at {0} and our conclusion holds. So we may assume {f(0^+) <f(0) =h(0)}. But in this case {l(\gamma) = \frac{h(\gamma x)-h(0)}{\gamma} = \frac{f(\gamma)-f(0)}{\gamma}} is going to be negative infinity as {\gamma \downarrow 0}. Recall from inequality (5), we have

\displaystyle \phi'(s)l(\gamma )\geq \langle{g},{x}\rangle

where {s} is between {h(0)} and {f(\gamma )=h(\gamma x)}. We claim that as {\gamma\downarrow 0}, {\phi'(s)} approaches a positive number. If this claim is true, then from the above inequality, we will have

\displaystyle -\infty \geq \langle{g},{x}\rangle

which cannot hold. Thus we must have {f} right continuous at {0}.

Finally, we prove our claim that {\phi'(s)} is approaching a positive number if {f(0^+) <f(0)}. Using mean value theorem, we have for some {s_0 \in [ f(0^+), f(0)]}

\displaystyle \begin{aligned} \phi(s_0)(f(0^+)-f(0)) & = \phi(f(0^+)) -\phi(f(0))\\ & = \lim_{ \gamma \downarrow 0 } \phi(f(\gamma))-\phi(f(0))\\ & = \lim_{\gamma \downarrow 0}\phi(s)(f(\gamma)-f(0))\\ & = (f(0^+)-f(0))\lim_{\gamma\downarrow 0}\phi(s). \end{aligned} \ \ \ \ \ (9)

Now cancel the term {f(0^+) -f(0)<0} above, we see that {\phi(s_0) = \lim_{\gamma\downarrow}\phi(s)}. We claim {\phi(s_0)>0}. If {\phi(s_0)=0}, then because {\phi} is increasing, we have that {\phi} is constant in {[f(0^+),f(0)]} as {\phi(f(0^+)) -\phi(f(0)) = \phi(s_0)(f(0^+)-f(0)) =0}. This contradicts our assumption that {\phi'(f(0))>0} and our proof is complete. \Box

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