When we have a convex function we know that a local minimum is a global minimum, and similarly for a concave function. What are some other situations where finding local extrema can yield global extrema with certainty? Are they common? Thanks!
2026-04-07 11:14:04.1775560444
Local Extrema and Global Extrema
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Two common classes of functions with similar relation between local and global are quasiconvex functions and their counterparts, quasiconcave functions.
Here, quasiconvexity is understood as the property that the set $\{x:\phi(x)< a\}$ is convex for every $a$. (In PDE theory this term has a different meaning). The composition of a convex function and an increasing function is quasiconvex.
Wikipedia gives a few examples such as this one, which looks like a translate of $\sqrt{|x|}$.