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Talk:Convex conjugate

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Clarification for a notation[edit]

What is the definition of the term ${\displaystyle \mathbb {R} _{--}}$ used in the table of convex conjugates (cases 6 and 8) in the column of the ${\displaystyle \operatorname {dom} (f^{\ast })}$?Kellertuer (talk) 08:57, 31 January 2014 (UTC)[reply]

Typo in rule for translation[edit]

What is y in the rule for the conjugate of f(x+b)? Is it x*? — Preceding unsigned comment added by 193.157.253.66 (talk) 08:21, 23 April 2012 (UTC)[reply]

Fixed. Zfeinst (talk) 15:49, 23 April 2012 (UTC)[reply]

Scaling properties[edit]

What does this (from the "Scaling properties" section) mean?

In case of an additional parameter (α, say) moreover

${\displaystyle f_{\alpha }($x)=-f_{\alpha }({\tilde {x}}),}

where ${\displaystyle {\tilde {x}}}$ is chosen to be the maximizing argument.

JadeNB (talk) 19:17, 23 March 2011 (UTC)[reply]

It is stated that ${\displaystyle f^{**}}$ is always convex and also always dominated by ${\displaystyle f}$. On the other hand, it seems to me that a concave function going to ${\displaystyle -\infty }$ cannot dominate any convex function, unless the dominated function is the constant function of value ${\displaystyle -\infty }$. But this is not allowed by the definition of ${\displaystyle {\overline {R}}}$ and, in turn, of ${\displaystyle f^{*}}$ and ${\displaystyle f^{**}}$. (Notably, this definition does not agree with the definition in the entry of fr.wikipedia, where $f^*$ and $f^{**}$ are allowed to take value ${\displaystyle -\infty }$.) I don't see a way out of this paradox. Delio.mugnolo (talk) 21:24, 2 December 2015 (UTC)[reply]

The lead text says that this is a generalisation of the Legendre transform, but looking at the Legendre transformation's page it's difficult to see in what the generalisation actually is; modulo differences in notation they seem pretty much the same, but I'm probably missing something. It would help a lot if someone with the required knowledge would write a brief paragraph about the differences between the two. Nathaniel Virgo (talk) 07:22, 10 February 2015 (UTC)[reply]

It appears to me as if the Legendre transform page is incorrect. The Legendre transform is for convex differentiable functions only and is defined by ${\displaystyle f^{*}(x^{*})=\langle x^{*},x\rangle -f($x)} where ${\displaystyle x}$ is such that ${\displaystyle \nabla f($x)=x^{*}} . The convex conjugate of a convex differentiable function coincides with the Legendre transformation, but as the convex conjugate can be defined for all functions it is more general. See page 94 of Boyd and Vandenbergheor[1]. Zfeinst (talk) 08:00, 10 February 2015 (UTC)[reply]