1

So, the problem is simple:

$\min_{\begin{array}{c}
                              x\\ 
                               y
                             \end{array}}f$

how to adjust the space between the two subscripts?

1

Using a stackengine stack, the vertical gap distances can be set. Differing horizontal alignments can also be set (default centered), if the stacking elements are of different width.

Furthermore, unlike using an array, the depth of the subscript is compatible with \min_x as shown in the last item.

\documentclass{article}
\usepackage{stackengine}
\stackMath
\begin{document}
\
$\min_{\stackunder[6pt]{\scriptstyle x}{\scriptstyle y}}f\quad
\min_{\stackunder[2pt]{\scriptstyle x}{\scriptstyle y}}f\quad
\min_x
$
\end{document}

enter image description here

2

I'd say that ehsanks now deleted comment about \substack is the most traditional. It only require amsmath and is shorter to write

\[
    min_{\substack{i,j \\ j<i }} ....
\]

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