what's the meaning for the upper tilt symbol in convex optimization for?

Question

I am kind of confused by the h with upper tilt symbol here: property

Thank you!

Answer 1

If, by "h with an upper tilt", you mean $h'()$, then that notation simply means the derivative of $h()$.

If you mean $\tilde{h}(x)$, the symbol is called a "tilde." This notation isn't completely standard but is used by some authors in convex optimization to mean the extension of a function with a limited domain to all of $R^{n}$ by defining the function to be $+\infty$ if $x$ is outside of the domain. If the domain of $h$ is $D$, then in this notation:

$\tilde{h}(x)=\left\{ \begin{array}{ll} h(x) & x \in D \\ +\infty & x \notin D \end{array} \right.$

Hopefully, the author of these slides defined the notation that they were using early in the presentation.