Easy interpretation of matrix multiplication with a set

Question

I have just started learning convex optimization. I am having little bit difficulties in some notations.

Currently I just encountered the following equation: $$ \boldsymbol{epi}(wf) = \left[ \begin{array}{ccc} I & 0 \\ 0 & w \end{array} \right]\boldsymbol{epi}(f) $$

where $w$ is positive and $f$ is convex. I have seen other matrix multiplications, but never encountered a matrix multiplication with a set, also how can I interpret it? Geometric interpretation would be helpful. Thanks.

Book reference page 79, 3.2.1 Nonnegative weighted sums

Answer 1

The notation is quite fine. Let $X$ be a set and $f:X\rightarrow D$ a function defined on $X$. Then \begin{equation} f(X):=\{f(x)\in D: x\in X\}. \end{equation} To the proof: Suppose $f:D\rightarrow \mathbb{R}$. The epigraph for $f$ is defined as \begin{equation} \operatorname{Epi}(f):=\{(x,a)\in D\times\mathbb{R}: f(x)\leq a\}. \end{equation} Therefore, for $w\in\mathbb{R}_{>0}$ \begin{equation} \operatorname{Epi}(w\cdot f):=\{(x,a)\in D\times\mathbb{R}: w\cdot f(x)\leq a\}. \end{equation} Now, suppose we have $(x,a)\in \operatorname{Epi}(f),$ i.e. $f(x)\leq a$. Then \begin{equation} \begin{pmatrix} \mathbf{1}&0\\0&w \end{pmatrix}\begin{pmatrix}x\\a\end{pmatrix}=\begin{pmatrix} x\\wa \end{pmatrix}, \end{equation} meaning $(x,wa)\in \operatorname{Epi}(w\cdot f)$.

Answer 2

The case $w = 0$ is obvious, and thus obmitted. For $w > 0$, let $D_w$ be the $(n+1)$-by-$(n+1)$ diagonal matrix with diagonal entries $\underbrace{1,\ldots,1}_{n \text{ times}}$, $w$ respectively. Then, $$\text{epi}(wf) := \{(z, t) \in \mathbb R^{n+1} | wf(z) \le t\} = \{(z, w\tau) | z \in \mathbb R^n, t\in \mathbb R, f(z) \le \tau\} = \{D_w[z\;\tau]^T | (z, \tau) \in \text{epi}(f)\} = D_w\text{epi}(f),$$ as claimed.