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In convex optimization,

$$ g=\underset{y\in C}{\mathop{\sup }}\,f(x,y) $$

where $f$ is convex in $x$ for each $y$, $y$ is belong to $C$.

we know that $g(x)$ is convex in $x$

I have two questions which are associated each other

  1. what should $C$(set of $y$) satisfy condition for $g$ is convex of $x$

  2. In case of $g=\underset{y\in \{y\left| y\le h(x)\} \right.}{\mathop{\sup }}\,\,\,f(x,y)$ , this $g$ is also convex????

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    Did you reread your question after you posted it? Please be sure your markup is correct, use LaTeX formatting (easy to Google), and proofread. I am barely able to read through this! – 2017-01-10
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    sorry about that, I edited that, thank you for pointing out to me. – 2017-01-10

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The set $C$ may not depend on $x$. Otherwise, even for a discrete set $C$, the function $h$ is convex since the supremum of convex functions is convex and that for fixed $y$, the function $h_y(x) = f(x,y)$ is convex.

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    Thanks for answer! ,So, even If $y=\{y\left| y=h(x) \right.\}$ , $g(x)$ is convex? for example $f(x,y)=x*y$ at that time, I think $g=\underset{y\in \{y\left| =h(x)\} \right.}{\mathop{\sup }}\,f(x,y)=xh(x)$ so I think that it depends on form of $h(x)$. – 2017-01-11
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    My very bad. The set $C$ may not depend on $x$. – 2017-01-11
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    Could I ask any reference about "The set $C$ may not depend on $x$?? – 2017-01-16
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    If the set $C$ does not depend on $x$, the collection of functions that you take the maximum over is fixed. The maximum of a (fixed) set of convex functions is convex, see [this book](http://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf), section 3.2.3. – 2017-01-16