TITLE: Difference between majorant, minorant and convex envelopes?
QUESTION [0 upvotes]: I have been reading "Convex Analysis and Global Optimization 2016" a Book by Hoang Tuy and stumble upon these terms: majorant, minorant. 
Does this concave majorant have the simirlar implication of the concave upper bound or something ?
Furthermore, the authors seem to use the term covex minorant and concave majorant together, does this mean they present the same geometric structure ?

REPLY [1 votes]: A function $m$ is a minorant (resp. majorant) of $f$ if, for every $x$ in your space $\mathcal{X}$, $m(x)\leq f(x)$ (resp. $m(x)\geq f(x)$).
It is common to find a minorant of a convex function, since many convex functions are not bounded above. These notions frequently appear in MM algorithms. Also, if you restrict to the set of affine minorants, this is closely related to notions of the subgradient of $f$.
The convex envelope of $f$, usually denoted $f^{**}$ is produced by taking the Legendre dual twice. This function is always a convex, lower-semicontinuous minorant of $f$, even if $f$ was not originally convex. Some view the convex envelope as the "closest" function to $f$ which is also convex and lower-semicontinuous.
If you multiply everything by $-1$, the geometry is almost analogous for concave majorants.
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