Subdifferential of $a^\text{T}x+\alpha\sqrt{x^\text{T}Bx}$ at $x=0$

I would like to compute the subdifferential of the function

$$f(x)=a^\text{T}x+\alpha\sqrt{x^\text{T}Bx}$$ where $\alpha>0$ and $B$ is symmetric positive definite.

Attempted Solution (I am brand new to subdifferentiability)

Since subderivatives, like normal derivatives, are linear, we can compute term-by-term. Since the first term is differentiable, we're really interested in computing the subdifferential of

$$\sqrt{x^\text{T}Bx}$$

at $x=0$. This has been asked in the question here, and I gather that the subdifferential should be

$$\{z:\sqrt{z^\text{T}{B}z}\leqslant\lambda_\text{min}\}$$

where $\lambda_\text{min}$ is the smallest eigenvalue of $B$. All together, the subdifferential is

$$\{a+\alpha z:\sqrt{z^\text{T}Bz}\leqslant\lambda_\text{min}\}$$

Is this correct? If so, can anyone elucidate why the subdifferential of $\sqrt{z^\text{T}Bz}$ is what it is? The linked question confused me.

2 Answers

The subdifferential of $f$ at $x$ is the set of $g$ that satisfy $f(y)-f(x) \ge g^T(y-x)$ for all $y$ in the domain. For convex functions, this is the set of supporting hyperplanes to $\operatorname{epi} f$ at $x$.

In the following, $\overline{B}(0,1)$ denotes the closed Euclidean unit ball.

Note that if $f$ is a norm, we have $f(0) = 0$.

Note for the Euclidean norm that $\|y\|_2 \ge g^T y$ for all $y$ iff $\|g\|_2 \le 1$. If you look at the epigraph of the norm you will see a satisfying geometric explanation.

Since $B$ is symmetric positive definite, it has a square root $\sqrt{B}$, and we can write $\sqrt{ x^T B x} = \|\sqrt{B} x\|_2$. Let me write $n(x) = \sqrt{ x^T B x}$ for convenience.

Then $n(y) = \sqrt{ y^T B y} = \|\sqrt{B} y\|_2 \ge g^T y = ((\sqrt{B})^{-1}g)^T \sqrt{B}y$, so we see that $n(y) \ge g^T y$ for all $y$ iff $\|(\sqrt{B})^{-1}g \| \le 1$. Or, equivalently you can write this as $\sqrt{B} \, \overline{B}(0,1)$. That is, $\partial n(0) = \sqrt{B}\, \overline{B}(0,1)$.

It is straightforward to see that if $f$ has a subdifferential $\partial f(x)$ at $x$ then, if $\alpha >0$, we see that $\alpha f$ has a subdifferential $\alpha \partial f(x)$ at $x$

If $f$ is continuously differentiable at $x$, then ${\partial f(x)} = \{ \nabla f(x) \}$, so we see that if $f(x) = a^T x$, then $\partial f(x) = \{ a \}$.

Finally, if $f_k$ (a finite number of) have subdifferentials $\partial f_k(x)$ at $x$, then (I am assuming that the $f_k$ are finite valued everywhere) $\partial (\sum_k f_k)(x) = \sum_k \partial f_k(x)$.

Combining all of these, we get $\partial f(x) = \{a\}+ \alpha \sqrt{B} \, \overline{B}(0,1)$.

Alternative: Another more technical approach is to note that $f$ is continuously differentiable everywhere except at $x=0$, so we can compute $\partial f (x) = \operatorname{co}\{ \xi | \xi \text{ is a limit point of } \nabla f(x_k), \text{ where } x_k \to 0, x_k \neq 0 \}$.

If $x \neq 0$ we can compute $\nabla f(x) = a + \alpha {(\sqrt{B} x)^T \over \|\sqrt{B} x\|_2 } \sqrt{B}$, and it is straightforward to see that the collection of accumulation points (taken as $x \to 0$) are $\{a\}+ \alpha \sqrt{B} \, \overline{B}(0,1)$.

Elaboration: This is an marginally different derivation of the formula for $\partial n(0)$.

We are looking for $g$ that satisfy $n(y)-n(0) = n(y) \ge g^T y$ for all $y$.

Note that $\sqrt{B}$ is symmetric, and $n(y) = \sqrt{y^T B y } = \sqrt{y^T \sqrt{B}\sqrt{B} y } = \|\sqrt{B} y\|_2$

Since $\sqrt{B}$ is invertible, note that $\{y|y \in \mathbb{R}^n\} = \{ (\sqrt{B})^{-1} y | y \in \mathbb{R}^n \}$, so we can write the subgradient characterisation as $n((\sqrt{B})^{-1} y) \ge g^T (\sqrt{B})^{-1} y = ((\sqrt{B})^{-1} g)^T y$ for all $y$.

Substituting the previous expansion of $n(y)$ we get that $g$ is a subgradient of $n$ iff $\|y\|_2 \ge ((\sqrt{B})^{-1} g)^T y$ for all $y$.

From the $3$rd paragraph above, we see that $g$ is a subgradient iff $\|(\sqrt{B})^{-1} g\|_2 \le 1$.

Then $\partial n(0) = \{g | \|(\sqrt{B})^{-1} g\|_2 \le 1 \} = \{ \sqrt{B} g' | \|g'\|_2 \le 1 \} = \sqrt{B} \overline{B}(0,1)$.

• So for example, if $B$ is a diagonal matrix with strictly positive entries, then the sub-differential is a hyperellipse centered at $a$ with axis lengths $\alpha b_{ii}$ for each $i$, correct? Feb 20 '18 at 22:04
• I would say lengths $\alpha \sqrt{b_{ii}}$. Feb 20 '18 at 22:11
• For intuition, look at the cone formed by the epigraph of the Euclidean norm, and consider all hyperplanes that pass through the origin, but do not contain any interior point of the cone. Feb 20 '18 at 22:13
• Yes--typo regarding the square root--thank you. The geometric intuition is very beautiful and makes sense. What I'm working on now is seeing whether your result $\sqrt{B}\overline{B}(0,1)$ is equivalent to my set $\{z:\|B^{1/2}z\|_2\leqslant\lambda_\text{min}\}$. Feb 20 '18 at 22:23
• @DavidM.: You can assume that $\sqrt{B}$ is diagonal, since the Euclidean norm is rotation invariant. Something does not look right about the formulation in the previous comment. Feb 20 '18 at 22:28

Hint: Find matrix $P$ and $A$ such that $B = P A P^{T}$. (see here for details ) Then using this express the term $\sqrt{x^\text{T}Bx}$ in terms of Euclidean norm i.e,

$$\sqrt{x^\text{T}Bx} = \| (P A^{\frac{1}{2}} )^T x \|$$

Now use chain rule for subdifferentials. (note that the matrix $( P A^{\frac{1}{2}} )^T$ has full rank.

Note that $\partial \|.\| (0) =B$ (the unite closed ball)

• Thank you for introducing me to the chain rule for subdifferentials Feb 20 '18 at 22:25