# Notation for an optimization function that receives a vector of pairs

In my optimization problem there are elements consisting of a time and a value, i.e. $$(t_0, v_0)$$. These pairs are stored in a vector $$v = [(t_0, v_0), (t_1, v_1), ... , (t_n, v_n)]$$. The vector $$v$$ serves as an input to an optimization function $$f$$ that computes a cost value from the input.

Is it mathematically common to say that $$V$$ is the set of all possible variants of $$v$$ and thus define $$f$$ as $$f : V \rightarrow \mathbb{R}$$?

• Since no one else has answered, I'd say it looks alright to me. Though I would probably say $f:\mathbb{R}^{n \times 2} \rightarrow \mathbb{R}$, if indeed $t_i \in \mathbb{R}$ and $v_i \in \mathbb{R}$. Jun 17 at 9:31

• Rearrange the pairs in a $$(n+1)\times 2$$ matrix $$M = \begin{bmatrix} t_0 & v_0\\ \vdots & \vdots\\ t_n & v_n \end{bmatrix},$$ so that you can replace $$V$$ with $$\mathbb{R}^{(n+1)\times 2}$$, as suggested by user @Nachiket in a comment.
• Avoid using the letter $$v$$ for this matrix, because it conflicts with the common notation that $$v$$ is the vector with entries $$v_1,v_2,\dots,v_n$$.