How important is to use the $\forall$ notation in the optimization frameworks?

Question

Consider I write the following optimization in my manuscript: $$\min_{X,Y} f(X,Y)\\s.t. ~x_{ij},y_{ij} \in \mathbb{R}^+,$$ where $X,Y$ are matrices, and by those constraints I mean their entries should be non-negative.

1. Is my notation regarding describing the constraints mathematically correct?
2. Do I need to add $\forall ij$ term to the constraint, and in that case, should I specify the range of $i,j$ if the dimensions of $X,Y$ are already defined somewhere before in the text?

Answer 1

This is probably just fine in the context you describe. Rather than asking about what's correct or important, put yourself in the position of your readers and think about what they would need to understand. In general, the less cluttered the symbols the better. You could consider replacing the $x$ in the first line by $X$ and the second line by

where $X$ and $Y$ have nonnegative entries.

Answer 2

I am more concerned about

1. $Y$ not included under the 'min',
2. the case insensitivity ($X$ vs $x$),
3. the $+$ occuring in superscript, and
4. the dimensions not being mentioned.

The following two notations seem better to me: $$\min_{X \in \mathbb{R}^{m_1 \times n_1}_+,Y \in \mathbb{R}^{m_2 \times n_2}_+} f(X,Y),$$ or $$\min_{X,Y} f(X,Y) \\ \text{s.t.} ~X \in \mathbb{R}^{m_1 \times n_1}_+,Y \in \mathbb{R}^{m_2 \times n_2}_+$$