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$$ f(x) = \begin{cases} 0 &\text{ if } |u| \le 1 \\ |u|-1 &\text{ if } |u| \gt 1 \end{cases} $$ $$ g(x) = \begin{cases} u^2 &\text{ if } |u| \leqslant 1 \\ 2|u|-1 &\text{ if } |u| > 1 \end{cases}$$

$$f^*(s) = \sup \left( \langle s,x\rangle -f(x)\right)$$

I am studying this for the first time. To learn from easier examples, Please help me to find the conjugate functions of $f(x),g(x)$

Thank you.

Pedro
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fordicus
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  • You really need to read about how to use the cases environment! Please take moment to see the TeX edits. – Pedro Jun 03 '17 at 16:54
  • Good. I hope someone could let me get through it smoothly then :) – fordicus Jun 03 '17 at 18:31
  • Thanks Pedro. It is resolved at the moment. I will try to learn more on TeX. – fordicus Jun 03 '17 at 21:30
  • In your case, you have $f(x) = \max(0, |x| - 1)$ and $g(x) = \max(2|x| - 1, x^2)$. Using the above, you can re-formulate the supremum computation, in both cases, as easy to solve convex optimization problems. I believe you should try it first, before looking at a complete answer. – Alex Shtoff Jun 04 '17 at 08:02

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