What's the appropriate terminology to indicate functions convex only in certain arguments? For example, consider the following function.
$f: X \times Y \rightarrow \mathbb{R}$, $f(x, \cdot)$ is convex.
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Does this answer your question? Convex function wrt. an argument – Mark S. Jan 11 '21 at 13:00
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@MarkS. I saw it and understood the notion but the answer does not tell us the terminology for such functions. – Jinrae Kim Jan 11 '21 at 13:40
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The question author used "convex wrt. its second argument" and the accepted answer used "convex w.r.t. $w$". So "$f$ is convex with respect to its second argument (for all $x\in X$)" or "$f$ is convex with respect to $y$" would be appropriate for your example. – Mark S. Jan 11 '21 at 13:44
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Actually, I googled papers with keyword "convex w.r.t. certain argument" but I can find only few papers. Anyway, thanks for answering :) @MarkS. – Jinrae Kim Jan 12 '21 at 06:13