Matrix convexity with a matrix-valued function

Question

According to the textbook Convex Optimization, a matrix function of the form

$$f(X) = - \log \det \left( - \left(X^{T}AX+B^{T}X+X^{T}B+C \right) \right)$$

is convex on

$$\boldsymbol{\mathrm{dom}}f = \left\{ X \in \boldsymbol{\mathrm{R}}^{m\times n} \mid (X^{T}AX+B^{T}X+X^{T}B+C)<0 \right\}$$

when $A \ge 0$, where $A \in \boldsymbol{\mathrm{S}}^{m}$, $B \in \boldsymbol{\mathrm{R}}^{m \times n}$, and $C \in \boldsymbol{\mathrm{S}}^{n}$.

However, how can I determine the convexity of $f(g(x))$ when $X$ has a matrix-valued function of $X=g(x)$, $g: \boldsymbol{\mathrm{R}}^{k}\to\boldsymbol{\mathrm{R}}^{m\times n}$. For example, if

$$X = \begin{bmatrix} h_0(x) & h_1(x) & \cdots & h_{n-1}(x) \\ h_n(x) & h_{n+1}(x) & \cdots & h_{2n-1}(x) \\ \vdots & \vdots & \ddots & \vdots\\ h_{(m-1)n}(x) & h_{(m-1)n+1}(x) & \cdots & h_{mn-1}(x) \\ \end{bmatrix} $$

what properties the scalar functions $h_i$ should have and what constraint the vector $x$ should have that can maintain the convexity of $f(g(x))$.

I'm a newbie of convex optimization so if my question has any ambiguity please let me know, thank you very much!

Answer 1

I think what follows should make your life easier:

• Fact: The function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is convex if and only if $g:\mathbb{R}\rightarrow\mathbb{R}, t\mapsto f(x+ty)$ is convex on $\text{dom}g:=\{t\in\mathbb{R}:x+ty\in\text{dom}f\}$ for any $x\in\text{dom}f$ and $y\in\mathbb{R}^n$.

Basically by applying this fact you should get a lovely $C^2$ function on the real line. Then all you need to do is second derivative test, and of course, you need to remember your linear algebra when taking the derivative.